Engineering Physics PDF

Turku University of Applied Sciences Engineering physics A short summary of physics for ICT Jaakko Lamminpää ver.2.3. Introduction This short book is for the engineering physics course in ICT. It provides some basic information about the topics, but more examples are discussed during the less...

Turku University of Applied Sciences Engineering physics A short summary of physics for ICT Jaakko Lamminpää ver.2.3. Introduction This short book is for the engineering physics course in ICT. It provides some basic information about the topics, but more examples are discussed during the lessons. Similarly theory is expanded during lessons in some topics and this book is not meant to be the only source of knowledge for the course. If you wish for more reading, I advise checking openstax AP College Physics and Physics classroom. On YouTube I recommend The Organic Chemistry Tutor and Khan Academy. Just remember, reading or listening by itself is a poor substitute for solving problems and doing math yourself. One of the pitfalls is that students tend to use the internet for pre-chewed answers. The final formula is there ready without any need for understanding or explanations. It may apply in one situation but not in another. It is advised to think what you are actually doing, and how you could get solve the problem yourself. One of the most often asked questions is how do I pass easily? The answer is simple, but boring. You need some time, calculator, a pen and paper. The most efficient way, and therefore the easiest, to learn is to actually solve problems yourself. You don’t need to solve everything, but that is the best way to learn the basics. Understanding also takes time and patience. One way to prepare for lessons, is to check or briefly read the material before lessons. Most important thing is to be active. If you don’t understand or know something it is always advised to look for answers. Better to ask questions and learn than sit in silence. Contents 1 SI – International system of units...................................................................... 4 1.1 Units...................................................................................................................................... 4 1.2 Prefixes.................................................................................................................................. 5 2 Kinematics....................................................................................................... 8 2.1 Displacement, time, and average velocity.............................................................................. 8 2.2 Average acceleration............................................................................................................. 9 2.3 Equation for uniform motion............................................................................................... 10 2.4 Graphs, velocity, acceleration and slope.............................................................................. 10 2.5 Motion in 2-dimensions....................................................................................................... 11 3 Mechanics...................................................................................................... 13 3.1 Definiton of a force.............................................................................................................. 13 3.2 Free body diagrams............................................................................................................. 14 3.3 Common type of forces....................................................................................................... 15 3.4 Newton’s laws..................................................................................................................... 17 3.5 Applying Newton’s laws..................................................................................................... 18 3.6 Momentum.......................................................................................................................... 20 4 Energy............................................................................................................ 21 4.1 Work.................................................................................................................................... 21 4.2 Kinetic and potential energy................................................................................................ 23 4.3 Conservation of energy........................................................................................................ 24 4.4 Power and efficiency........................................................................................................... 25 5 Static electricity.............................................................................................. 27 5.1 Charge and conservation of charge...................................................................................... 27 5.2 Coulomb’s law..................................................................................................................... 28 5.3 Electric fields....................................................................................................................... 28 6 DC-circuits..................................................................................................... 33 6.1 Resistance and Ohm’s law................................................................................................... 33 6.2 Kirchoff’s laws.................................................................................................................... 34 6.3 DC circuit analysis............................................................................................................... 35 7 Electromagnetism........................................................................................... 36 7.1 Magnetism is related to current and vice versa.................................................................... 36 7.2 Charged particle in a magnetic field.................................................................................... 37 7.3 Magnetic field of a conductor.............................................................................................. 39 8 Electromagnetic induction and AC.................................................................. 40 8.1 Faraday’s law and Lenz’s law............................................................................................. 40 8.2 Applications......................................................................................................................... 41 8.3 Alternating current............................................................................................................... 44 3 1 SI – International system of units Consider a situation where you have to measure the length of your room without a tape measure. How would you do it? Perhaps you could lay down and see that it is excatly three times as long as you. But another person might get a different result when measuring your room. There is a clear need for a standardized measurement system. Most common and widely used is the international system of units, abbreviated as SI. In short, formulas in this course expect that you use SI units. While some exceptions exist, a safe way is to convert everything into SI units before calculating. 1.1 Units SI-system has base units and derived units. The base units are listed below. Note that sometimes the quantity symbol for length might vary and that mass has a prefix kilo. This is discussed more later. Table 1. Basic SI quantities and units Physical quantity Symbol Unit name Symbol Time t second s Length l metre m Mass m kilogram kg Electric current I ampere A Temperature T Kelvin K Amount of substance n mole mol Luminous intensity l candela cd SI has a multitude of derived units which have a specific name. Below is listed some of the most used ones and some of these are used in this course. These are standardized and can be used as is in calculations. However, you can always substitute the derived unit with SI units to check if the formula or calculation yields the correct units. 4  Table 2. Derived SI quantities and units Physical quantity Symbol Unit name Symbol In SI base units -1 Frequency f Hertz Hz s -2 Force F Newton N kg m s 2 Pressure p Pascal Pa Nm 2 -2 Energy, work, heat E, W, Q Joule J kg m s 2 -3 Power P Watt W kg m s Electrical charge Q Coulomb C As 2 -3 2 Electrical potential U,V Volt V kg m s A Also many known physical quantities have a combination of base units. Such as 2 3 3 velocity (m/s), acceleration (m/s ) volume (m ), and density (kg/m ). There is no need to memorize all the derived units. These are introduced better when they appear later in this course and they come more familiar through exercises and calculations. Note that when calculating with SI units you can trust that the end result will be in SI units. Meaning that even if you are uncertain about the unit of mass times acceleration, knowing that this is how force is calculated, this must yield to Newtons – the derived unit for force in SI system. 1.2 Prefixes To have an understanding of the magnitude, the SI system uses metric prefixes. These are essentially just “10 to the power something” -type of factors. For example, 1 km 3 means 1 thousand meters. Here k stands for kilo, which is 10 and you can simply 3 replace the letter k with 10. Similarly you have 1 millimeter, which is one thousandth -3 of a meter. You can substitute the milli with 10. Example 420 𝑚𝑚 = 420 ⋅ 10−3 𝑚 = 0.42 𝑚 0.0052 𝑚 = 5.2 ⋅ 10−3 𝑚 = 5.2 𝑚𝑚 Note: using powers of 10 is an easy way to think of this. It is dividing or multiplying with factors of 10, causing the decimal separator to move to the 3 -6 left or right. 10 = move three to right. 10 = move six to left. 5 In calculations these prefixes should always be marked with the units. The SI standard requires the use of base units to achieve the derived units. This means changing prefixes to powers of 10 when doing calculations. Example Resistance is calculated by dividing the voltage with current. If the voltage is 5 V and current is 10 mA, then 𝑈 5𝑉 5𝑉 𝑉 𝑅= = = = 500 = 500 Ω 𝐼 10 𝑚𝐴 0.01 𝐴 𝐴 If not converted to amperes, you would get wrong results. 𝑈 5𝑉 𝑉 𝑅= = = 0.5 ≠ 0.5 Ω 𝐼 10 𝑚𝐴 𝑚𝐴 While in some cases units with prefixes might cancel one another out, it is adviced that unless you are certain of the units cancelling out, you should change the prefixes into standard SI units. The one notable difference is the SI unit for mass – the kilogram (kg). It includes the prefix kilo in the standard units and should be used as such. Meaning that grams are changed into kilograms. In what way should you write your results? Usually prefixes or powers of 10 are used. For example, instead of 100 000 m it is more understandable to write 100 km. With some larger numbers the powers of 10 are more favourable. For example, the charge of an electron is 1.6 ⋅ 10−19 𝐶. However, these are just guidelines and I personally don’t really care as long as they are correct and easy to understand, unless a specific way is requested. But for calculations it is beneficial to learn to use the powers of 10 instead of long numbers, like 1.2 ⋅ 10−8 𝑚 instead of 0.000000012 𝑚, because those are easily written and read incorrectly. For engineers, the unit prefixes provide a way to evaluate answers easily and when calculating current one should have an understanding what could be expected values. Depending on the situations it could be milliamperes or amperes or even microamperes, but having kiloamperes is extremely rare. Similary calculating the length of the cable leading to results like nanometers doesn’t really make sense. 6  Table 3. Prefixes Prefix Symbol Factor Power tera T 1 000 000 000 000 1012 giga G 1 000 000 000 109 mega M 1 000 000 106 kilo k 1 000 103 hecto h 100 102 deca da 10 101 (none) (none) 1 100 deci d 0.1 10−1 centi c 0.01 10−2 milli m 0.001 10−3 micro µ 0.000001 10−6 nano n 0.000000001 10−9 pico p 0.000000000001 10−12 7 2 Kinematics Kinematics is the part of physics that deals with motion. The forces that cause the motion are discussed later in chapter 3. There are four relevant quantities for motion: displacement (or distance), time, velocity, and acceleration. These concepts should be familiar to most from everyday life even if the calculations and formulas are not. In this chapter we first look at simple motion without acceleration, then with acceleration and finally expand this from 1 dimensional motion (moving in a single line) to 2 dimensions (projectile motion or moving on a map). 2.1 Displacement, time, and average velocity. In continunuous motion we can describe the movement of an object with a single mathematical function or a formula. This includes constant motion as well as having a constant acceleration. In non-continunuous motion the theory is not much more difficult, but the movement just needs to be divided into different sections, like driving with a constant velocity for some time and then decelerating to a stop. In physics, we have two distinct terms for length in kinematics: displacement and distance. Displacement means how far we are from a starting point. Distance means the total length of the path we took. For example, the straight line distance from my home to a lake is 5 km but when I walk along the road it is actually 7.5 km because the road is not a straight line. If I choose to take a more scenic route the distance is even more. These also relate to concepts of velocity and speed. Velocity is related to displacement while speed is related to the distance. Moreover, straight line has a direction and therefore velocity is a vector with a direction while speed is a scalar. The problem arises that these are used interchangeably in everyday life. In physics these sometimes have a difference. For example, if you are programming a CNC router (computer controlled drill or a robot) to carve something, the path is quite different from the displacement and in coding you should differentiate which one is needed. Figure 1. Displacement and distance. Walking from home to a lake. Path A is 7.5 km and path B is 8.2 km. These differ one from another and also from the displacement itself. 8  I assume that time and length are quite familiar, so how to calculate an average speed or velocity of an object? Think about the units of velocity. They are usually km/h, m/s or even miles per hour. The unit itself already tells us that it is a length divided by time. More accurately it is the change in displacement over a certain period of time. Δ𝑥 𝑥2 − 𝑥1 𝑣= = (1) Δ𝑡 𝑡2 − 𝑡1 The Δ (=delta) means change and is important when defining velocity more accurately and discussing graphs later in this chapter. To exact, this is the average velocity during time Δ𝑡. For simplicity, this formula can be written 𝑥 𝑣= 𝑡 if the starting point 𝑥1 = 0 and starting time 𝑡1 = 0. Symbol x is often used for displacement and s for distance to distinguish between velocity and speed. However, you are fine with using just one of the two. Just make sure you know which one you are using. The difference between the velocity and speed can best be described with an example. Most notable situation is when you return to starting point. Then the displacement is 0 and therefore the velocity is 0, no matter how long a path you took and with what speed. With mathematics you can solve the time it takes to travel a known distance with known speed from this equation. You can do the same for the distance as well. So knowing basic maths means you have less to remember. 2.2 Average acceleration From everyday life you know that if you accelerate your car, you will be going faster. And if you decelerate, you will slow down. In physics, the acceleration is presented as the change of velocity over time. Δ𝑣 𝑎= (2) Δ𝑡 The Δ (=delta) means change and is important distinction from average velocity v and emphasizes the change from one velocity to another. To be more accurate, this is the average acceleration and is suitable for most physics problems, but in some cases we might need to look at instantaneous acceleration. 9 2.3 Equation for uniform motion If you have an object that is moving with an original velocity 𝑣0 and it is experiencing constant acceleration (or deceleration) 𝑎, the displacement for the object can be calculated 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 (3) 2 This can also be applied when there is no original displacement (𝑥0 = 0) or original velocity (𝑣0 = 0) or acceleration (𝑎 = 0). This is one of the most useful equation when it comes to motion. 2.4 Graphs, velocity, acceleration and slope The average velocity and acceleration are useful, but in some cases we are interested in the instantaneous velocity and acceleration. For example, when we are driving a car our velocity changes depending on the speed limit and traffic. How do we define the velocity at a specific moment during acceleration? This can be best explained with a graph. In constant velocity the displacement is changing with a constant rate while in acceleration the velocity is increasing and the displacement is changing faster. Figure 2. Left graphs shows a constant velocity. Right graph is showing an increase in velocity. A rate of change in math is also called a derivative or the slope. Meaning that you can calculate the velocity at any given time by just drawing a tangent in (x,t)-graph and calculating the slope of the tangent. This is also why it was important to keep the Δ in the equation 1. If the velocity is change of displacement over change of time, this directly corresponds the slope of the (t,x)-graph, even the units match. Or to be more accurate, they have to match. In figure 2 for the left graph the slope is constant and fairly easy to calculate. For the right one we would need to draw some helping lines. 10  Example A car is moving according to the graph. Calculate the velocity when 𝑡 = 11.0 𝑠. Note that the red lines are drawn as a part of the solution. The instantaneous velocity can be calculated as the slope of a tangent (red line) of the graph where 𝑡 = 11.0 𝑠 (blue line). Red dashed lines are to help read values for displacement and time. Δ𝑥 𝑥2 − 𝑥1 4.0 𝑚 − 1.0 𝑚 3.0 𝑚 𝑚 𝑣= = = = = 0.25 Δ𝑡 𝑡2 − 𝑡1 18.0 𝑠 − 6.0 𝑠 12.0 𝑠 𝑠 The slope also give a visual cue. The steeper the slope, the higher the value – meaning that you can evaluate the velocity just by looking how steep the slope is. Negative slope means moving backwards and flat one means staying still. Similarly for instantaneous acceleration, you can determine the acceleration from (t,v)-graph as a slope. This can again be deduced from the units. If x-axis is time and y-axis velocity, acceleration being change of velocity divided by change of time must corresponds to the slope. 2.5 Motion in 2-dimensions The previous chapter discussed movement on a single line or in 1-dimension. While the velocity is still considered a vector, the direction comes actually meaningful in more dimensions. Two simple examples are velocities that are not parallel and projectile motion where there is acceleration in y-direction while the x-velocity remains constant. When calculating with vectors, you should always draw the situation. When you divide velocities and/or accelerations in x- and y-components, you form right-angled triangles. For these you can apply Pythagoras’ theorem and trigonometry. For projectile motion, one can assume that the motion in x-direction is uniform. Therefore 11 the displacement 𝑥 = 𝑣𝑥 𝑡, where 𝑣𝑥 = 𝑣 cos 𝛼. The time of flight is controlled by the motion in y-direction. While there is no one equation, you can derive what is needed by 1 combining two equations: 𝑦 = 𝑦0 + 𝑣0𝑦 𝑡 − 𝑔𝑡 2 and we know that at any given time 2 the velocity is 𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡. Most notable situation being the peak, when 𝑣𝑦 = 0 and the time is half of the total flight. Example You are rowing a boat 2.0 m/s. The river is flowing left to right 0.50 m/s. In what direction you need to row, to cross the river directly (shortest path). How fast is the crossing when width of the river is 35 meters? If the river is moving you to the right, you need to have an opposing component to prevent you from moving. Meaning that your rowing velocity’s x-component must be equal but opposite to the river flow. The 𝑣 = 2.0 𝑚/𝑠 and 𝑣𝑥 = 1.0 𝑚/𝑠. with trigonometry The time depends on the 𝑣𝑦. With 𝛼 𝑣𝑦 𝑣𝑥 cos 𝛼 = sin 𝛼 = 𝑣 𝑣 𝑣𝑦 = 𝑣 cos 𝛼 1.0 𝑚/𝑠 Time it took sin 𝛼 = 2.0 𝑚/𝑠 𝑠 = 𝑣𝑦 𝑡 𝛼 = 30 ° 𝑠 𝑠 𝑡= = 𝑣𝑦 𝑣 cos 𝛼 35 𝑚 = 𝑚 2.0 ⋅ cos 30 ° 𝑠 ≈ 20 𝑠 You could also calculate the length you would need to go 30 degrees left and use 𝑣 instead of 𝑣𝑦. Or you could calculate the angle between the rowing velocity and 𝑣𝑥. 12 3 Mechanics Mechanics is basically calculating forces which is closely related to acceleration, and therefore, into kinematics. This section assumes that you have understood the basics from the previous chapter. In the real world, problems of this nature concern both – the mechanics and the kinematics. For ICT this chapter focuses on Newton’s laws and their basic applications. 3.1 Definiton of a force In layman’s terms a force is anything you would call a push or a pull. Generally, a force is any influence that causes a body to accelerate. Forces on a body can also cause stress in that body, which can result in the body deforming or breaking. Though forces can come from a variety of sources, there are three distinguishing features to every force. These features are the magnitude of the force, the direction of the force, and the point of application of the force. Forces are often represented as vectors and each of these features can be determined from a vector representation of the forces on the body. The magnitude of force is measured in units of mass times length over time squared. The derived SI unit is the Newton (N), where one Newton is one kilogram times one meter over one second squared. This means that a force of one Newton would cause a one-kilogram object to accelerate at a rate of one meter per second squared. kg⋅m [F]=1 N=1 2 s In addition to having magnitudes, forces also have directions. As we said before, a force is any influence that causes a body to accelerate. Since acceleration has a specific direction, force also has a specific direction that matches this acceleration. The direction of the force is indicated in diagrams by the direction of the vector representing the force. Figure 3. Force vector can be given as the magnitude and angle, or as the components. 13 3.2 Free body diagrams A free body diagram (also called force diagram) is a tool used to solve engineering mechanics problems. As the name suggests, the purpose of the diagram is to "free" the body from all other objects and surfaces around it so that it can be studied in isolation. We will also draw in any forces or moments acting on the body, including those forces and moments exerted by the surrounding bodies and surfaces that we removed. It is an excellent tool to evaluate the situation and the net force. It also helps to keep track of all the forces and directions in calculations. As we do not discuss moments, we are not using them in this book. In general, moment exists when the point of application is not the centre of mass and it causes rotation in the body. The first step in solving most mechanics problems will be to construct a free body diagram. This simplified diagram will allow us to more easily write out the equilibrium equations for statics or strengths of materials problems, or the equations of motion for dynamics problems. Free body diagrams consist of: A simplified version of the body (often a dot or a box) Forces shown as straight arrows pointing in the direction they act on the body Moments are shown as curves with an arrow head or a vector with two arrow heads pointing in the direction they act on the body One or more reference coordinate systems By convention, reactions to applied forces are shown with hash marks through the stem of the vector Figure 4. A problem illustrated as forces acting on a body. If needed, you should also include velocities and accelerations and their respective directions. 14 3.3 Common type of forces Gravitational Force. Bodies with mass attract other bodies with mass. On Earth, the mass of Earth is pulling everyone towards its centre. The gravitational force on a body 2 is called weight and it is presented as 9.81 m/s times the mass of the object resulting in Newtons as the unit (compare to Newton’s second law). 𝐺̅ = 𝑚𝑔 (4) Figure 5. Gravitational forces always act downward on the center of mass (COM). Normal Forces (or Reaction Forces): Every object in direct contact with the body will exert a normal force on that body which prevents the two objects from occupying the same space at the same time. Note that only objects in direct contact can exert normal forces on the body. An object in contact with another object or surface will experience a normal force that is perpendicular (normal) to the surfaces in contact. It the force preventing you from falling through the floor. Figure 6. Normal forces always act perpendicular to the surfaces in contact. The barrel in the hand truck shown on the left has a normal force at each contact point. 15  Friction Forces: Objects in direct contact with the body can also exert friction forces, which will resist the two bodies sliding against one another, on the body. These forces will always be perpendicular to the surfaces in contact. Friction force depends on the materials between surfaces and the normal force between the objects. For slippery objects we can assume that the coefficient of friction is zero and there is no friction force. Note that the direction of the 𝐹𝜇 is different from the normal force. 𝐹𝜇 = 𝜇𝑁 (5) Tension in Cables: Cables, wires or ropes attached to the body will exert a tension force on the body in the direction of the cable. These forces will always pull on the body, as ropes, cables and other flexible tethers cannot be used for pushing. Figure 7. The tension force in cables always acts along the direction of the cable and will always be a pulling force. The forces presented are the most common, but other forces such as pressure from fluids, spring forces and magnetic forces exist and may act on the body. 16 3.4 Newton’s laws 3.4.1 Newton’s first law Newton's first law states that: ‘A body at rest will remain at rest unless acted on by an unbalanced force. A body in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force.’ This law, also sometimes called the law of inertia, means that bodies maintain their current velocity unless a force is applied to change that velocity. If an object is at rest with zero velocity it will remain at rest until some force begins to change that velocity, and if an object is moving at a set speed and in a set direction it will remain at that same velocity until some force acts on it to change its velocity. It is important to note that the net force is what will cause acceleration and therefore the change in velocity. The net force is the sum of all forces acting on the body. For example, we can imagine gently pushing on a large rock and observing that the rock does not move. This is because we will have a friction force equal in magnitude and opposite in direction opposing our gentle pushing force. The sum of these two forces will be equal to zero, therefore the net force is zero and the change in velocity is zero. It is important to notice that in real life there are always multiple forces acting on a body. There is always friction and air resistance, but sometimes these can be negligible. This simplifies the problem but has to be done with the effects in mind. You can neglect friction when skating on ice but not when dragging a box on gravel. 3.4.2 Newton’s second law Newton's second law states that: ‘When a net force acts on any body with mass, it produces an acceleration of that body. The net force will be equal to the mass of the body times the acceleration of the body.’ Σ𝐹̅ = 𝑚𝑎̅ (6) Σ means the sum, and in this case the sum of forces. The force and acceleration are vectors meaning that you need to take directions into account when doing calculations. Usually this means dividing the forces into x- and y-components and calculating those separately. Based on the above equation, you can infer that the magnitude of the net force acting on the body will be equal to the mass of the body times the magnitude of the acceleration, and that the direction of the net force on the body will be equal to the direction of the acceleration of the body. If the body is at rest or moving at a constant velocity, the acceleration is zero. This then presents the first law as a formula Σ𝐹̅ = 0. 17 3.4.3 Newton’s third law Newton's Third Law states that: ‘For any action, there is an equal and opposite reaction.’ By ‘action’ Newton meant a force, so for every force one body exerts on another body, that second body exerts a force of equal magnitude but opposite direction back on the first body. Since all forces are exerted by bodies (either directly or indirectly), all forces come in pairs, one acting on each of the bodies interacting. Figure 8. The gravitational pull of the Earth and Moon represent a Newton's Third Law pair. The Earth exerts a gravitational pull on the Moon, and the Moon exerts an equal and opposite pull on the Earth. Image adapted from Public Domain images, no authors listed. Though there may be two equal and opposite forces acting on a single body, it is important to remember that for each of the forces a Third Law pair acts on a separate body. This can sometimes be confusing when there are multiple Third Law pairs at work. Therefore it is important to draw free body diagrams and consider multiple objects and acting forces separately. For example, if you are pushing a wall, the wall is experiencing that force but you are simultaneously experiencing equal but opposite force from the wall. If you are drawing a free body diagram of the wall, then it does not include the force the wall is applying to you. 3.5 Applying Newton’s laws Applying Newton’s laws is straightforward if you follow the steps and include all the forces correctly. Then it only comes down the one’s ability to do math. Everything is centered around the free body diagram and Newton’s second law. It is important to note, that the second law also includes the first law if the object is stationary or has a constant velocity. If the problem is in 2-dimensions, you need to divide the second law for x- and y-directions separately – the steps and laws are still the same. More examples are presented in class. Best way to learn these is by doing these – a lot. 18 Steps for solving a problem 1. Draw a free body diagram. Include all the forces and name them appropriately Choose which direction is positive Mark down the direction of acceleration and velocity nd 2. Write down Newton’s 2 law If in 2D, write the equation for x-direction and y-direction separately (note that the object can be stationary in one direction but still have acceleration in the other) If there is acceleration Σ𝐹 = 𝑚𝑎 If stationary or constant speed Σ𝐹 = 0 3. Solve for the wanted variable Identify which variable from the equations you need to solve If you don’t know all the other variables in that equation solve them from other equations Example A lamp (mass 1.50 kg) is hanging from the ceiling with one wire. What is the tension in the wire? We start by drawing a free body diagram. There is only components in y-direction. Newton’s second law states that Σ𝐹̅ = 𝑚𝑎̅ Because the object is at rest 𝑎 = 0 and Σ𝐹̅ = 0 T−G=0 T=𝐺 = 𝑚𝑔 𝑚 = 1.50 𝑘𝑔 ⋅ 9.81 ≈ 14.7 𝑁 𝑠2 It is important to choose positive and negative directions and assign these to forces. Now if the answer would be negative, the force would be pointing downwards. 19 3.6 Momentum In this chapter we discuss solely about linear momentum, but it is good to know that also rotational momentum exists and it works in similarly, just with slightly different formulas for rotating bodies. The most important application of momentum in coding is in game design and collisions. Basically every interaction with another object is a collision; car crashing a tree, the character walking into NPCs, or even a butterfly landing on a flower – the scale is just different. Games sometimes omit actual momentum with simple velocity based movement, but this can lead into all sort of bugs. Therefore it is good to understand the basics of momentum. Momentum is closely related to force and inertia. If two objects are moving with the same speed which one is easier to stop – the one which weighs less or the one which weighs more? Answer is obvious and this is due to momentum and it is defined as 𝑝̅ = 𝑚𝑣̅ (7) meaning that if objects weigh the same, the faster one is more difficult to stop. As vector, the direction needs to be included also and plays a crucial role in conservation of momentum. In collisions the momentum is always conserved. This means that the net momentum (or sum of momentums) objects have before a collision must equal the net momentum after the collision. For two objects this means 𝑝̅1𝑖 + 𝑝̅2𝑖 = 𝑝̅1𝑓 + 𝑝̅2𝑓 (8) 3.6.1 Impulse To change momentum one must exert force. However, time must also be considered. For example, Superman can stop a train by pushing it, but sliding backwards at the same time. This gives him more time and requires less force to bring the train to a stop without any harm. On the other hand, if he was to stop the train in an instant a much larger force would be required and train would be crushed. The relationship of force and impulse can be derived from Newton’s second law. Remembering that acceleration is change of velocity over change of time, we get Δ𝑝 𝐹= (9) Δ𝑡 An example of the use of this in coding is present in class. 20 4 Energy Energy itself has many forms, like work, kinetic energy, potential energy, or thermal energy (heat). Basic thing to understand beforehand is that energy is energy and while some forms seem different, they are still the same. Different forms are discussed in different chapters but they still relate to one another. This is all due to conservation of energy which states that energy cannot disappear and it cannot be created from nothing. It can only change from one form to another. 4.1 Work To make sense of conservation of energy, we need to understand the concepts of work and energy. Work in general is a force exerted over a distance. If we imagine a single, constant force pushing a body in a single direction over some distance, the work done by that force would be equal to the magnitude of that force times the distance the body traveled. If we have a force that is opposing the travel (such as friction), it would be negative work. 𝑊 =𝐹⋅𝑑 (10) Figure 9. In instances with a constant force and a constant direction, the work done to a body will be equal to the magnitude of that force times the distance the body travels. For forces opposing the motion, the work will be negative. For instances where forces and the direction of travel do not match, the component of the force in the direction of travel is the only piece of the force that will do work. In these cases, you need to find the component of the force that parallel to the direction of the movement. By this logic we can also state that forces perpendicular to travel don’t 21 do any work. In Figure 10, the force doing work is calculated 𝐹𝑥 = 𝐹𝑝𝑢𝑠ℎ ⋅ cos 𝜃. Sometimes we can also calculate the net work, which means that then we only consider the net force in our calculation. Figure 10. Only the components of a force in the direction of travel exert work on a body. Forces perpendicular to the direction of travel will exert no work on the body. In the case of a force that does not remain constant, we will need to account for the changing force. To do this we will integrate the force function over the distance traveled by the body. Just as before, only the component of the force in the direction of travel will count towards the work done, and forces opposing travel will be negative work. This might seem complex, but the idea is to show that many of physics’ formulas are either integrals or derivates, and the formulas we often use are special cases where we either calculate the average value or assume that some of the variables are constant. 𝑥2 𝑊 = ∫ 𝐹(𝑥) 𝑑𝑥 𝑥1 22 4.2 Kinetic and potential energy When discussing energy, we will usually break energy down into kinetic energy and potential energy. Kinetic energy is the energy of a mass in motion, while potential energy represents the energy that is stored up due to the position. In its equation form, the kinetic energy of an object is represented by one half of the mass of the body times its velocity squared. If we wish to determine the change in kinetic energy. 1 𝐸𝐾 = 𝑚𝑣 2 (11) 2 As a note, a body that is rotating will also have rotational kinetic energy, but this is not discussed in this course. Potential energy, unlike kinetic energy, is not really energy. Instead, it represents the work that a given force will potentially do between two instants in time. Potential energy can come in many forms, but we will focus only on the potential energy. This represent the work that the gravitational force will do. We often use the potential energy in place of the work done by gravity. The change in gravitational potential energy for any system is represented by the product of the mass of the body, the value 2 g (9.81 m/s ), and the vertical change in height between the start position and the end position. This can be also derived by calculating the work done when lifting an object. 𝐸𝑃 = 𝑚𝑔ℎ (12) Figure 11. When finding the change in gravitational potential energy, we multiply the mass by g (giving us the weight of the object) and then multiply that by the change in the height of the object, regardless of the path taken. 23 4.3 Conservation of energy The concepts of work and energy provide the basis for solving a variety of kinetics problems. Generally, this method is called the Conservation of Energy, and it can be boiled down to the idea that the work done to a body will be equal to the change in energy of that body. By dividing energy into kinetic and potential energy pieces as we often do in dynamics problems, we arrive at the following base equation for the conservation of energy. 𝑊 = Δ𝐸𝑃 + Δ𝐸𝐾 (13) If compared to the equation presented in class, you need to understand that it is the same – just a different expression. This perhaps helps to understand that if energy is added to the system as work, it is positive. If system loses energy, it the work is negative. It is important to notice that unlike Newton's Second Law, the above equation is not a vector equation. It does not need to be broken down into components which can simplify the process. However, we only have a single equation and therefore can only solve for a single unknown, which can limit the method. The figure 12 illustrates how a skateboarder perceives the conservation of energy. When standing at the top, the skateboarder has only potential energy. As he goes down the potential energy is turning into kinetic energy and he gains speed. At the bottom he has no potential energy and the velocity, and therefore the kinetic energy, reaches its maximum value. Then he starts to turn the kinetic energy into potential energy when going up the ramp on the other side. Even if energy is always conserved, this does not continue forever. Friction will do work that moves the energy from the system into the surroundings as heat and noise causing the skateboarder’s total energy to decline and finally he becomes to a stop at the bottom. Figure 12. The conservation of mechanical energy illustrated in a ramp. 24 4.4 Power and efficiency Related to the concepts of work and energy are the concepts of power and efficiency. At its core, power is the rate at which work is being done, and efficiency is the percentage of useful work or power that is transferred from the input to the output of some system. Average power is defined as the work done in a certain amount of time. Power at any instant is the derivative of work with respect to time. Work can be defined in many different ways through various forms of energy as presented in the previous chapters. 𝑊 𝑃= (14) 𝑡 or for instantaneous power 𝑑𝑊 𝑃= 𝑑𝑡 The unit of power is watts for the metric system, where one watt is defined as one joule per second, or one Newton-meter per second. Maximum power ratings are often a primary specification for motors and engines, but also important for ICT in the field of electronics. Powerhandling and maximum power ratings in electronics are discussed more in chapter 6. Figure 13. Assuming the two cars above have the same mass, it would take the same amount of work to get them up to a set speed (such as 100 km/h). However, the more powerful car would be able to get to this speed in a much shorter time period. Any devices with work/power inputs and outputs will have some loss of work or power between that input and output, due to things like friction. While energy is always conserved, some energies such as heat may not be considered useful. A measure of the useful work or power that makes it from the input of a device to the output is the efficiency. Specifically, efficiency is defined as a portion of input energy that a machine can utilize. For example, if machine takes in 100 J but can use only 60 J 25 for working, that means that the efficiency is 0.60. In other words, it is the percentage of energy it can use to work. With power being the work over time, efficiency can also be described as power out divided by the power in to a device (the time term would cancel out, leaving us with our original definition). 𝑊𝑜𝑢𝑡 𝑃𝑜𝑢𝑡 𝜂= = (15) 𝑊𝑖𝑛 𝑃𝑖𝑛 It is impossible to have efficiencies greater than one (or 100%) because that would be a violation of the conservation of energy; however, for most devices we wish to get the efficiencies as close to one as possible. This is not only because greater efficiencies waste less work/power, but also because any work or power that is "lost" in the device will be turned into heat that may build up. In computers, the excessive heat causes the processor units to run slower. 26 5 Static electricity Static electricity means usually stationary charges and how those interact with one another through electric fields. It also includes static situations and movement caused by the forces described for example by Coulomb’s law. 5.1 Charge and conservation of charge An atom is constructed from electrons, protons and neutrons. Electrons are negatively and protons positively charged and have the magnitude of one elementary charge. 𝑞 = 1.602 ⋅ 10−19 C If an object has an excessive amount of either of the charges, this causes the object itself to be charged either positively or negatively. It is important to note that both the elementary charges and normally charged objects have a rather low charge, usually no larger than nano- or microcoulombs. Charging happens usually either by contact or rubbing between two surfaces causing electrons to move from one object to another. This can also be achieved with electric potentials. The protons are relatively stable and in most cases it is electrons moving. However, objects or molecules can be positively charged if they have lost their electrons. Charges do not disappear, because that would require the electrons or protons themselves disappear. The charge is in system is always conserved, meaning that the charges can still move from one place to another, but the overall amount stays the same. This means if two objects are neutral but after charging one has a +1.0 nC charge, the other must have -1.0 nC. Materials can be divided into two categories; conductors and insulators. Conductors usually have plenty of free electrons and they are good charge carries, meaning that they conduct electricity well. These include for example copper, gold, and most metals. Insulators on the other hand don’t allow the charges to move as freely and they are bad at conducting electricity. Insulators include for example plastic, ceramics and most non-metals. 27 5.2 Coulomb’s law Electrostatic forces interact between two charged objects; like charges repel while unlike charges attract. For example, two negative charges repel while a positive and a negative charge would attract one another. Coulomb’s law calculates the magnitude of the forces between two point charges. |𝑞1 𝑞2 | 𝐹=𝑘 (16) 𝑟2 where 𝑞1 and 𝑞2 are the charges of the two point charges separated by the distance 𝑟. The constant 𝑘 ≈ 9.0 ⋅ 109 Nm2 /C2. This is for a vacuum and is relatively same in the air. If the medium is something that does not allow the electric field (and therefore the force) permeate through, the value of 𝑘 is different. For example, a plastic separating two charges causes the force to be weaker. The force itself always interacts along the line between the two charges. Figure 14. The magnitude of the electrostatic force 𝐹 between point charges 𝑞1 and 𝑞2 separated by a distance 𝑟 is given by Coulomb’s law. Note that Newton’s third law (every force exerted creates an equal and opposite force) applies as usual—the force on 𝑞1 is equal in magnitude and opposite in direction to the force it exerts on 𝑞2. 5.3 Electric fields A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding the earth (and all other masses) represents the gravitational force that would be experienced if another mass were placed at a given point within the field. In the same way, the Coulomb force field surrounding any charge extends throughout space. One way of thinking the effects of a charge 𝑄 is to have a ‘test charge’ 𝑞 and think what would happen to it in different places around the actual charge. 28  To simplify things, we would prefer to have a field that depends only on Q and not on the test charge. The electric field is defined in such a manner that it represents only the charge creating it and is unique at every point in space. Specifically, the electric field 𝑬 is defined to be the ratio of the Coulomb force to the test charge: 𝐹 𝐸= (17) 𝑞 Unit for electric field is 𝐍/𝐂 (newtons per coulomb). But as we later see, it can also be presented in 𝐕/𝐦 (volts per meter). According to Coulomb’s law we can calculate the electric field for a point charge: |𝑄| 𝐸=𝑘 (18) 𝑟2 The electric field is thus seen to depend only on the charge Q and the distance r; it is completely independent of the test charge q. 5.3.1 Field lines for particles and uniform fields Drawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. The electric field, like all vectors, can be represented by an arrow that has length to its magnitude pointing to its direction. Note that the electric field is defined for a positive test charge; the field lines point away from a positive charge and toward a negative charge. Where would a proton go? The electric field strength is proportional to the number of field lines per unit area. This pictorial representation, in which field lines represent the direction and their closeness represents strength, is used for all fields: electrostatic, magnetic, and others. Figure 15. The electric field surrounding three different point charges. a) A positive charge. b) A negative charge of equal magnitude. c) A larger negative charge. 29  If there are other particles causing their own electric fields, the electric field 𝐸 can always be calculated as a sum of the electric fields. The fields can be sketched by thinking where the net force acting on a positive charge would be pointing. Note that drawn lines can never cross. Figure 16. The electric field of a) two like charges and b) two unlike charges. Uniform electric field is an electric fields which has the same magnitude and direction at any place. This means that the force acting on a charged particle remains constant. Uniform field is usually achieved by placing two parallel charged plates near each other (see figure 17). Figure 17. Two charged plates create a uniform electric field. Note that at the ends the field is not totally homogenenous, but as an approximation it can be thought as such. 30 5.3.2 Potential energy, Potential and voltage If you lift a particle in an electric field, you do work and can calculate the amount of energy needed. We can neglect the gravity if the weight is notably smaller than the force applied by the electric field – which is usually the case. Is you lift a particle (charge Q) against a uniform field you would do work (same as potential energy) which is equal to force and displacement against the field (see figure 18.) 𝑊 =𝐹⋅𝑥 𝐸𝑃 = 𝑄𝐸𝑥 (19) The force comes from the formula (18). In similar fashion one can calculate the work done by the field if it moves a particle. Figure 18. Work done if a positive charge is lifted from the negative plate. The electric potential is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. Notice that this is different from potential energy. In this case, there is no actual charge in the field but instead we imagine a charge and calculate the potential energy per charge. This gives us 𝐸𝑝 𝑉= 𝑄 = 𝐸𝑥 While this is good to know, more important is the potential difference which relates to something more familiar. The potential difference is called the voltage. In a way voltage is can be seen as a result of the forces pulling a charged particle or the work electric field can do. 𝑈 = 𝑉𝐵 − 𝑉𝐴 = 𝐸𝑑 (20) 31 where E is the electric field and d is the distance between two potentials. This gives us an alternative to calculate the electric field. While the units are different, they are the same magnitude, meaning that 𝑁/𝐶 = 𝑉/𝑚. 𝑈 𝐸= (20) 𝑑 5.3.3 Electrostatic applications One application is painting or printing by charging particles. For example, the paint is charged negatively while a metallic part is put in a positive potential. This causes the metal to pull negative charges, or in this case the paint, towards it. In medicine development, the charging can be used to separate and combine different drug materials by charging them. Static charging and uniform fields can also be used to create electron or any other charged beams. Charges are accelerated in a homogeneous field which does work and gives them kinetic energy. This relates back to conservation of energy. In similar fashion, an electric field can be used to alter the path of the beam. If the field is perpendicular, it causes a force and therefore acceleration that redirects the beam. This nd relates back to Newtons 2 law and kinematics. 𝐸𝑘 = 𝑞𝑈 Figure 19. Electric field used to accelerate an electron and then the path is changed with a perpendicular electric field Conservation of energy gives that you can calculate the kinetic energy as the work done by the field. The perpendicular field causes an electron to be pulled towards the positive plate (see figure 19). You can calculate the acceleration with Newton’s second law if you know the force applied by the electric field. This acceleration affects only the duration the particle is in the field, and it is only dependent on the velocity in x- direction. The field does not change that because it only affects in y-direction. If you calculate the time the acceleration is applied, you can calculate the velocity it receives in y-direction. There is no example calculation for this, because you have these in your homework and I want you to do them with thinking like this, not just taking a ready formula. 32 6 DC-circuits The DC stands for direct current. Meaning that we are discussing electric circuits where the current and the voltage do not alternate. The physics is quite simple on a broad level. If there is a potential difference (voltage) the positive charges are pulled to the lower potential and negative charges are pulled to the higher potential. In a uniform field these would correspond to the positive and negative plates, and in a normal battery these are simply the positive and negative terminals. We have already defined the voltage. The current is defined as the electrons movement, basically how much charge is flowing per second. In circuits it is important to distinguish between traditional current flow and electron flow. Before knowing that it was the electrons that were moving, the traditional current flow was chosen from positive to negative. The electron flow is actually what is happening. For calculations we use the traditional current flow. The electrons cannot move totally freely. As with static electricity we noted that some materials conduct electricity better than others. The resistivity is a fundamental property of a material that measures how strongly it resists electric current. However, depending on the shape of the objects, the same material can be a better or worse conductor for current. Therefore, in circuit we usually have resistors which have a resistance. The resistance measures how strongly a certain object (not material) resists current flow. 6.1 Resistance and Ohm’s law Circuits have two types of connections; components can be in series on in parallel. Components connected in series are connected along a single "electrical path", and each component has the same current through it, equal to the current through the network. Components connected in parallel are connected along multiple paths, and each component has the same voltage across it, equal to the voltage across the network. Resistance of multiple resistors connected in series can be calculated as the sum of all resistances 𝑅𝑇 = 𝑅1 + 𝑅2 + ⋯ + 𝑅𝑛 (21) The resistance of multiple parallel resistors can be calculated from 1 1 1 1 = + +⋯+ (22) 𝑅𝑇 𝑅1 𝑅2 𝑅𝑛 33  Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. The constant showing the magnitude of this relationship is the resistance. Ohm’s law can be formulated as 𝑈 = 𝑅𝐼 (22) (I use letter U is used to differentiate from potential V). It is important to note that this applies to one resistor as well as multiple resistors. The difference then being that you have the voltage over a set of resistors, the total resistance and the total current. 6.2 Kirchoff’s laws Kirchoff’s current law states that, for any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be formulated as: Σ𝐼 = 0 Kirchoff’s voltage law states that: The directed sum of the potential differences (voltages) around any closed loop is zero. Σ𝑉 = 0 The voltage law can also be used to observe how the potential changes in the circuit. The resistors cause voltage drops while the voltage source provides (adds) voltage. The voltage drop can be calculated with Ohm’s law. But it is important to note, that if you are calculating a loop where you go against the current, the voltage over the resistor is positive (see figure. This is because when you are going against the current, you are actually moving to higher potential. Figure 20. When applying voltage law, the sign depends on the direction which we are going and the direction of the current or the cell. 34 6.3 DC circuit analysis There are multiple approaches, and all have their advantages and disadvantages depending on the circuit and what needs to be solved. Unfortunately it is impossible to give a one solution. Finding the best way, and learning to ‘see’ it, is the result of practicing it yourself multiple times. Luckily, you can solve it in many different ways, so finding the optimal solution in the beginning is not important. Basically most approaches are just different ways of applying Kirchoff’s laws and therefore instead of individually explaining them, I just encourage on using the Kirchoff’s laws. There are few steps that you should still do before starting: 1. Draw the schematic if needed 2. Write down the symbols for components as well as the values 3. Mark down the direction of the currents (guess if you don’t know) From this you can first see what can be calculated with Ohm’s law. You can get some voltage drops, currents or resistances even without applying Kirchoff’s laws. Note that if you apply the Ohm’s law for multiple resistors, you get the total resistance, total voltage over the resistors and the total current in the circuit. This might not be enough to give values for individual branches or components. An example is provided during class. Writing one with step-by-step solutions here takes too much time and space. You can check examples from YouTube or even from Wikipedia. 35 7 Electromagnetism Magnetism and electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, and it is responsible for electromagnetic radiation such as light. Basically the magnetism and electricity are closely related to one another. The physics helps us to understand for example how alternating current is formed and how or why there is interference in electrical circuits. 7.1 Magnetism is related to current and vice versa All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole will point toward the north. The two poles are thus named the north magnetic pole and the south magnetic pole. Figure 21. a) What a compass would show in magnetic field. b) The magnetic field can be presented as a line from north to south. The strength of the field is proportional to the closeness (or density) of the lines. c) The field actually continues through the magnet. Only certain materials, such as iron, cobalt, nickel, and gadolinium, exhibit strong magnetic effects. Such materials are called ferromagnetic. A group of materials made from the alloys of the rare earth elements are also used as strong and permanent magnets; a popular one is neodymium. Other materials exhibit weak magnetic effects, which are detectable only with sensitive instruments. Not only do ferromagnetic materials respond strongly to magnets (the way iron is attracted to magnets), they can also be magnetized themselves — that is, they can be induced to be magnetic or made into permanent magnets. Other forms of magnetism are paramagnetism and diamagnetism. Paramagnetic materials are weakly attracted to magnets while diamagnetic materials are weakly repelled by magnets. 36  Electrical currents also cause magnetic fields. This can be used to create electromagnets. These are used from a wrecking yard crane that lifts scrapped cars to controlling the beam of a 90-km-circumference particle accelerator to the magnets in medical imaging machines. The electrical current in a wire creates a magnetc field which can be presented with a right hand rule. Thumb shows the direction of the current and curled fingers the direction of the magnetic field (see figure 22). Figure 22. a) The magnetic field of a circular current loop is similar to that of a bar magnet. b) A long and straight wire creates a field with magnetic field lines forming circular loops. c) When the wire is in the plane of the paper, the field is perpendicular to the paper. Note that the symbols used for the field pointing inward (like the tail of an arrow) and the field pointing outward (tip of an arrow). 7.2 Charged particle in a magnetic field What happens when we have a moving charges in magnetic field? The answer is related to the fact that all magnetism is caused by current, the flow of charge. If you have a moving charge in a magnetic field, the magnetic field exerts a force on moving charge. The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force 𝐹 on a charge 𝑞 moving at a speed 𝑣 in a magnetic field of strength (also called magnetic flux density) 𝐵 is given by 𝐹 = 𝑞𝑣𝐵 sin θ (23) where 𝜃 is the angle between the directions of 𝑣̅ and 𝐵̅. The optimal case is when 𝑣̅ is perpendicular to 𝐵̅, and the sine is for calculating the perpendicular component. In fact, this is how we define the magnetic field strength 𝐵 — in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength B is called the tesla (T). The direction of the force cannot be determined from this formula directly. 37  The direction of the magnetic force 𝐹 is perpendicular to the plane formed by 𝑣 and 𝐵, determined by the right hand rule (yes, this is another right hand rule. Just know which one is used for which). Note that the right hand rule applies to current or positive particle. For negative particle, the force is to the opposite direction. Figure 23. Left a presentation of the right hand rule for 𝐹 = 𝑞𝑣𝐵. All perpendicular to another. In the middle a force on a positive charge moving west. Right a picture how to use right hand rule for to find out that the force is down. This means that the force is always perpendicular to velocity and causes acceleration perpendicular to velocity. The acceleration changes the direction but the magnitude of the velocity, causing the particle in a circular path due to centripetal force. Figure 24. A negatively charged particle moves in the plane of the page in a region where the magnetic field is perpendicular into the page (represented by the small circles with x’s—like the tails of arrows). The magnetic force is perpendicular to the velocity, and so velocity changes in direction but not magnitude. Uniform circular motion results. 38 While we have not discussed circular motion, the simplified version is that the Newton’s second law can be used when the acceleration is considered centripetal acceleration. 𝑣2 𝐹 = 𝑚𝑎 = 𝑚 (24) 𝑟 In magnetic field, this force must equal to the magnetic force on a particle (see eq. 23). 𝑣2 𝑞𝑣𝐵 = 𝑚 𝑟 Which in turn can be used to solve physics problems, like the ones you have as homework. The important thing to understand that you have a magnetic force which causes a circular path and this can be expressed also through Newton’s second law. Similar kind of method is used for velocity selector, understanding Hall effect, and even when deriving induction voltage in later chapters. Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor (current) is transmitted to the conductor itself. Without further proofs, the force acting on a conductor in a magnetic field is 𝐹 = 𝐼𝑙𝐵 sin 𝜃 (25) 7.3 Magnetic field of a conductor As stated previously, magnetism and moving charges are related to one another. Magnetism is produced by moving charges. In atomic level it is the spin of electrons that makes certain materials magnetic. In larger scale it is a current in a wire that causes a magnetic field around it. This can be illustrated with a right hand rule (the other one). With this, you can figure out how a straight wire or a coil, or even a solenoid, would produce a magnetic field. When these fields are known we can also see how they would affect other wires carrying currents. Figure 25. a) how to use right hand rule to figure the direction of the magnetic field inside a solenoid. b) cross section diagram of the solenoid. 39 8 Electromagnetic induction and AC Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field. Electromotive force can in laymans terms be understood as voltage. Meaning, that by moving a magnet around a conductor or conductor around a magnet, you are able to generate a voltage. This is a really simplified explanation of how generators work. This principle can also be used in transformers to raise or lower the voltage. 8.1 Faraday’s law and Lenz’s law Faraday was first to show how induction worked by connecting a circuit to a metallic ring (see fig. 26). When the circuit is closed or opened, the galvanometer registers a current even if the circuit is not connected to the other. Figure 26. The Faraday’s experiment to illustrate how changing current changes magnetic field and induces a voltage on another circuit. The most common interpretation of Faradays law states that: The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path. Magnetic flux can be understood as magnetic lines going through a certain area and the are being limited by the enclosed path. Φ = 𝐵𝐴 cos 𝜃 , (26) where the B is the flux density and the A is the area. Cosine is used to calculate the component of B perpendicular to A. And with this we can calculate the induced voltage (or electromotive force / emf). ΔΦ 𝜖=− (27) Δ𝑡 This is also know as Faraday’s law of induction, but instead of just using the formula, it also good to know what it actually presents. 40  The Faraday’s law does not actually originally include anything about the direction of the induced current, but the minus sign in front of the formula is used to emphasize this. The direction of the induced current is defined by Lenz’s law, which states that: The current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force which opposes the motion. If you want to think this simply, things just want to be the way they used to be. So the induced current is opposing the change. By using the right hand rule, one can always deduce the direction of the induced current. If magnetic field is becoming weaker, the current provides backup by creating a field in the same direction. If the field is becoming stronger, the current creates an opposing magnetic field. 8.2 Applications For normal applications a single winding or a loop does not provide much voltage. However, stacking loops after another causes each of them to generate a voltage. This means that a coil with N windings generates a voltage induced by the change of the flux which can be expressed simply ΔΦ 𝜖 = −𝑁 (28) Δ𝑡 Basically generators, and most induction based applications, utilize coils with multiple loops to generate a larger induction voltage. Electric generators induce a voltage (emf) by rotating a coil in a magnetic field, a briefly mentioned in the beginning of the chapter. The induced voltage depends on the rate of change of magnetic flux. If you are rotating a coil (see fig. 27.), the flux does not change at a steady pace. The current, and the voltage, also change directions when the coil turns the other way around due to Lenz’s law. Instead of steady DC voltage, it resembles a sine curve. Without going too much into detail, this explains how alternating current (AC) is formed, why it has the shape of sine wave, and why it changes between a positive and negative voltages.The rotating coil (change of area) is not the only thing causing induction. One options is also the change of the magnetic flux density. This can be intentional or unwanted. Induction cooktops have electromagnets under their surface. The magnetic field is varied rapidly producing eddy currents in the base of the pot, causing the pot and its contents to increase in temperature. Induction cooktops have high efficiencies and good response times but the base of the pot needs to be ferromagnetic, iron or steel for induction to work. 41  Magnetic flux Φ = 𝐵𝐴 ⋅ cos 𝜃 The angle 𝜃 at any given time 𝜃 = 𝜔𝑡 The voltage any given instant 𝑑Φ 𝑈=− 𝑑𝑡 𝑑 = − (𝐵𝐴 ⋅ cos 𝜔𝑡) 𝑑𝑡 = −𝐵𝐴𝜔 ⋅ sin 𝜔𝑡 Figure 27. Operating principle of a generator and deriving the how the induced voltage is shaped like a sine wave If the magnetic field is changing and affecting a wire unwantedly, it causes interference. While small fluctuation in most circuits are meaningless, in some, the effects are notable. A common example is an amplifier where the AC signal meant to generate sound, receives noise caused by alternating magnetic fields. These disturbances are often caused by other AC currents. The changing current creates a changing magnetic field, which in turn affects the current in the signal path. This is the reason that some amplifiers have a 50 Hz hum – the same frequency as the AC coming from the wall outlet. How can you shield against this unwanted interference? One option is to keep the signal path and the cause of the interference separated. In circuit design, the power section is usually further away from the signal path. You can also use shielded cables. The cable has insulated conductor wrapped in a common conductive layer. The sield acts as a Faraday cage and prevents the interference. The noise signal is directed away by connecting the common layer to the ground. 42 8.2.1 Transformers Transformers transform voltages from one value to another. For example, laptops, power tools and small appliances have a transformer built into their plug-in unit that changes 230 VAC into whatever voltage the device uses. Transformers are also used at several points in the power distribution systems. Power is sent long distances at high voltages, because less current is required for a given amount of power, and this means less line loss. But high voltages pose greater hazards, so that transformers are employed to produce lower voltage at the user’s location. Transformers are based on Faraday’s law of induction. The two coils are called the primary and secondary coils (see figure 28.). In normal use, the input voltage is placed on the primary, and the secondary produces the transformed output voltage. Not only does the iron core trap the magnetic field created by the primary coil, its magnetization increases the field strength. Since the input voltage is AC, it induces an AC output voltage. Figure 28. Construction of a transformer where the magnetic field is strengthened by an iron core and the schematic symbol for a transformer. From the Faraday’s law we can derive that for a transformer the induced secondary voltage depends on the primary voltage and the ratio of loops. This is known as the transformer equation, and it simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils. 𝑉𝑠 𝑁𝑠 = (29) 𝑉𝑝 𝑁𝑝 If the resistance is negligible, the power of the input equals the power of the output. This is nearly true in practice—transformer efficiency often exceeds 99%. Equating the power input and rearranging some terms we get also that 𝑉𝑠 𝐼𝑝 = (30) 𝑉𝑝 𝐼𝑠 43 8.3 Alternating current In previous chapters we showed how induction occurs. When this is used in generators, the turning of the coil in a magnetic field leads to sine waveform current, and voltage. While one could use different kind of movement to create differently altering voltage, the method resulting in sine wave has become the standard. Mostly due to the simplicity of the system. Alternating current, and voltage, are both changing in magnitude and in direction. The values depend on the instance when they are measured. So how can we have meaningful information about the AC? The normal outlet provides 230 volts of alternating current. This is also called the effective voltage or the rms voltage (root mean square). It is basically showing a value at which the system provides voltage if it would correspond a DC voltage. Note that this is different from average voltage. For a traditional AC the average voltage would be zero as the wave is symmetric around the x axis. This is the most useful value for most applications. Same applies for the current. In some cases we can also use the peak voltage, or current. Peak voltage is the highest point or highest value of voltage for any voltage waveform. The link between the effective voltage and the peak voltage is well known. While it depends on the waveform, for the sine waves it is a factor of √2. 𝑉𝑝𝑒𝑎𝑘 𝑉𝑟𝑚𝑠 = (31) √2 The same can again be applied to the currents. 𝐼𝑝𝑒𝑎𝑘 𝐼𝑟𝑚𝑠 = (32) √2 Also one important factor for the AC is the frequency. In generators this comes from the speed at which the coil is turned. It corresponds on the period it takes for the coil to return to the same position. The normal household outlets here are standardized for 50 Hz. In other countries it may vary. Moreover, in signal processing we can also have any kinds of frequencies, even varying frequencies or waves that are a sum of many multiple waves. In its basic form an AC voltage can be depicted as a sine wave 𝑉 = 𝑉0 ⋅ sin(2𝜋𝑓𝑡) (33) 44 Attributions The physics notebook is based on the OpenStax AP College Physics which is freely available online and licensed under Creative Commons Attribution License v4.0 (CC BY 4.0). All material produced for this notebook can be used according to the same license. 45