Magnetic Force and Moving Charges

• These are study notes for AP Physics 2 about Magnetic Fields.

History of Magnetism

• Magnets were first discovered over 2000 years ago in China and Greece and were used for various non-scientific purposes.
• The Greeks coined the name "magnetite" because magnetic rocks were found in the province of Magnesia.
• Magnets were first used for navigation after 1000 A.D. by Chinese, European, and Persian mariners.
• Shaped as a needle and floated on water, magnetic material would point toward the north pole.
• The ability to tell which direction was north was a critical factor in ushering in the age of exploration.
• The phenomenon of magnetism was explained in 1600 by William Gilbert.

Magnet Properties

• Magnets have two ends (poles) called north and south.
• Opposite poles attract each other.
• Like poles repel each other.
• Attraction or repulsion between magnetic poles exerts the magnetic force.

Magnetic Poles

• When a magnet is cut in half, each piece still has a north and a south pole.
• Pieces will continue to have both poles no matter how many times the magnet is cut.
• Every magnet has both north and south poles.
• It is not possible to have only a "north magnet" or a "south magnet"
• Magnetic poles cannot be isolated

Magnetic Poles and Electric Charges

• Behavior of magnetic poles (north and south) are similar to electric charges (positive and negative).
• Opposite poles/charges attract and like poles/charges repel.
• Certain materials are naturally magnetic, while electrical properties result from physical rubbing.
• There are independent positive and negative charges, but magnetic materials always contain a north and a south pole.

Magnetic Fields

• Electric field lines showed how electric charges would exert forces on other charges.
• A similar concept is used in magnetism.
• Iron filings act as little bar magnets and align with the magnetic field of the large magnet.
• Magnetic field exits one end of the magnet and returns to the other end.
• Field lines extend through the magnet to make a complete loop.
• Magnetic field lines are defined as arbitrarily leaving the north pole and reentering at the south pole.
• They specify the direction that the north pole of another magnet in the field "wants" to point.
• The more lines per unit area, the stronger the field.
• Magnetic field lines form complete loops and go right through the magnet.
• Like electric fields, magnets will produce interesting magnetic fields depending on their configurations.
• Two magnets with their north poles next to each other repel each other.
• Two magnets with their opposite poles next to each other attract each other.

The Earth's Magnetic Field

• The Earth's magnetic field is similar to that of a bar magnet and is caused by the circulation of molten iron alloys in the Earth's outer core.
• The Earth's "north pole" is really a south magnetic pole because the north ends of magnets are attracted to it.
• Magnetic poles are not located exactly along the earth's axis of rotation but are close.
• The magnetic field extends from the core to the outer limits of the atmosphere (magnetosphere).
• The magnetosphere protects life on Earth from ionizing radiation that would cause great harm.
• Particles in the solar wind are prevented from entering the earth's atmosphere by the magnetosphere.
• Some particles enter the magnetosphere and are directed along the magnetic field lines to the magnetic north and south poles, where the fields are strongest.
• Passing through the atmosphere, they collide with oxygen and nitrogen molecules, transferring their energy to excite the molecules, which then re-emit the energy in the form of light.
• This kinetic energy to light energy transformation produces both the Aurora Borealis and Aurora Australis
• The Aurora Borealis is shown encircling the south magnetic pole (geographic north pole).

Origin of Magnetic Fields

• Each piece will still have a north and south pole when a magnet is cut in half and pieces are cut in half continued down to the atomic level.
• The real explanation of magnetism requires an understanding of quantum mechanics.
• Classical concepts allow for understanding of the phenomenon.
• In 1819, Hans Christian Oersted set up a current-carrying wire above a magnetized needle and discovered a relationship existed between electricity and magnetism.

Electric Currents Produce Magnetic Fields

• When there is no current in the wire, the needle aligns with the Earth's magnetic field and points north.
• When current was supplied, the magnetized needle rotated on the pivot and pointed at a right angle to the electric wire.
• The direction of rotation would depend on the direction of the current.
• Electric current produces a magnetic field affected the compass needle more strongly than the earth's magnetic field.
• Current carrying wires generate magnetic fields that deflect compass needles.
• With no current, compass needles all point north.
• When the current is turned on, the compass needles point in the direction of a circle around the wire.
• The Earth's magnetic field is caused by the circulation of molten alloys in the Earth's outer core.
• Oersted demonstrated that currents create magnetic fields.

Magnetic Field Origin (Bohr Model)

• Classically, the Bohr Model of the atom shows electrons orbiting the nucleus. moving charges generate a magnetic field.
• Electrons move around their atoms, so they generate magnetic fields.
• Most of these magnetic fields generated by electron movements can cancel each other.
• A nonzero net magnetic field will result at times.

Magnetic Field Origin (Electron Spin)

• Another contribution to the magnetic field of the atom - "electron spin" explained by quantum electrodynamics.
• The electron is a point particle but effect explained as if it really spins.
• This spin adds to the strength of the magnetic field created by the orbital motion of the electrons.

Magnetic Dipole

• Magnetic dipole is simply, a very tiny magnet.
• Some atoms have electron configurations that cause them to be magnetic dipoles.
• Electrons orbiting a nucleus and spinning on their axes constitute moving charges, and therefore create magnetic fields
• Effects may cancel out to create no net magnetic field.
• These atoms are permanent magnetic dipoles if their magnetic effects do not fully cancel out.

Magnetic Dipole Moment

• Magnetic dipole moment of a magnet or an atom or electron is pointed in the direction of the north pole of the object.
• Vector represents the strength of the magnetic field produced by the object and how strongly the dipole will interact with an external magnetic field.
• An electron is traveling clockwise (Using the Bohr model), which means we have a current going counterclockwise (conventional current is opposite electron flow).
• The vector represents the direction and strength of the magnetic dipole moment due to the electron movements.

Magnetic Domain

• Represented atoms and their magnetic fields with arrows (north and south pole).
• Each group of like-pointed arrows is called a magnetic domain.
• Each atom has its magnetic moment in the same direction within a magnetic domain.
• For a random group of atoms, there is no real reason for their magnetic poles to align.
• The middle picture shows a large field domain, which is more of a magnet on the diagram.
• Arrows are all pointing up in the third drawing, and it visualizes a bar magnet with the north pole up.

Types of Magnetic Materials

• Ferromagnetic materials can be permanently magnetized by the application of the field - domains will line up in the same direction ~ Examples include iron, nickel, and cobalt.
• Paramagnetic materials weakly interact with the external field and temporarily line up some of their domains; attracted to magnets.
• When the field is removed, the domains resume their random ordering examples include magnesium and lithium.
• Diamagnetic materials - domains line up opposite the external field, materials are repelled by a magnet.
• Example include mercury and bismuth.

Ferromagnetic Materials

• Paper clips are steel which is mainly iron (a ferromagnetic material)
• Magnetic domains in paper clip align in the presence of an external magnetic field.
• Each paper clip becomes a small magnet.

Example 1: Magnetic Domains

• Iron is ferromagnetic, so it's domains will line up with the applied external field of the magnet.
• Each iron paperclip will have its north pole pointed in the same direction, becoming a chain of little bar magnets and are attracted to each other.
• Effect is not permanent, so domains will fall out of alignment after the magnet is removed
• Copper is paramagnetic, and it's domains will not line up enough to be attracted to the magnet.

Summary

• A magnetic field is a vector field that describes the magnetic force exerted on moving charges, electric currents, or magnetic materials.
• Magnetic fields are created by magnetic dipoles or combinations of dipoles.
• Magnetic fields exist as closed loops through magnetic dipoles.
• Magnetic fields are not created by monopoles.
• Magnetic field lines form closed loops.
• Inside a magnet, magnetic field lines point from south to north.
• Outside a magnetic field, magnetic fields lines point from North to South
• Magnetic field lines allow to visualize magnetic field around a magnetic dipole or a magnetic object.
• Magnetic fields are created by magnetic dipoles and describe the direction of the magnetic force at various points in space.
• Magnetic permeability describes a substance's ability to form internal magnetic fields.
• Paramagnetism is weak attraction to a magnetic field that some materials experience due to magnetic dipole moments created from electrons' spin.
• Ferromagnetic substances have an inherent magnetic field due to the alignment of their magnetic domains.
• The net magnetic field through any closed surface will be zero.

Direction of a Magnetic Field due to a Current Carrying Wire

Electric Currents Produce Magnetic Fields

• Oersted observed that the direction of the magnetic field depends on the direction of the electric current.
• The direction of the field is given by the right-hand grip rule.
• Orient your right hand thumb in the direction of the current, and B field follows the path followed by your curled fingers.
• To find the direction of the magnetic field at any point on the concentric circles, draw a tangent to the circle.
• Tangent line points in the direction of the magnetic field at that point.
• there is no component of the magnetic field pointing towards or away from the wire.
• Grasp the pencil with your right hand and "curl" your fingers by wrapping them around the pencil.
• The curled fingers indicate the direction of the magnetic field.
• To the left of the wire, the magnetic field is coming out of the page.
• To the right of the wire, the magnetic field is going into the page.
• A magnetic field is needed to be shown as three dimensional to be understood as it encircles around current carrying wire
• Showing 3rd dimension on paper is the challenge
• Positive charge will feel an electric force in the electric direction
• Negative charge will feel electric force in opposite direction

Vector Symbols

• Picture a field line (a vector with magnitude & direction) like an arrow.
• The head of the arrow is the direction of the field.
• If the magnetic field is into the page, the tail of the arrow is seen (x).
• If the magnetic field is out of the page, the point of the arrow is seen (•).
• Using the x-dot notation, the magnetic field around a current carrying wire can be shown with Curl Right Hand Rule.
• The right-hand grip rule can be used on the magnetic field inside and outside a current carrying loop.

Example 1: Electric Currents Produce Magnetic Fields

• For an electric current flowing into the page, the magnetic field will be circles in clockwise direction.
• Orient your right hand thumb into the page, in the direction of the current.
• The magnetic field is in the same direction as your curling fingers and is directed in the clockwise direction.

Example 2: Electric Currents Produce Magnetic Fields

• For an electric current flowing to the right, the magnetic field around this current carrying wire is circles in counter-clockwise direction above the wire, and in clockwise direction when below it
• Orient your right hand thumb (pencil) to the right
• Curl your four fingers, trying to grab pencil
• Fingers will go into page below the current, and come out-of page above it, showing the mag. field direction

Example 3: Electric Currents Produce Magnetic Fields

• For an electric current flowing in counter-clockwise direction around a loop, the magnetic field is •, meaning out-of-the-page
• Use the right-hand grip rule
• Point your thumb toward the tangent line
• Fingers will curl out of the page in inside of the loop, and into the page on outside of the loop

Example 4 and 5

• Illustrate direction of magnetic field at a point and direction of magnetic compass for a nearby magnetic source e.g. current carrying wire
• A Magneto static Compass needle aligns with the local field (combined local fields at the point that are vector added)

Example 6: Magnetic Field Lines

• Illustrates addition of magnetic field vectors at a point which is influenced by multiple source currents and using Vector addition approach
• B is also a vector.

Electric Currents

Oersted Observed

• Oersted observed the direction of electrical current dictates the direction of Magnetic field around it in circles.
• The magnitude of this field at a distance ' r ' away from straight conducting wire carrying steady electric current I is given by:

Magnetic Field Units

• The symbol for the Magnetic Field is B, a vector with both magnitude and direction.
• The unit of B is the Tesla (T) measured in Newton/amp·meter.
• The Gauss (G) is also frequently by scientists, where 1 G = 10⁻⁴ T.
• To gain perspective, the magnetic field of the Earth at its surface is around 0.5 x 10⁻⁴ T (or simply 0.5 G).

Magnetic Field Due to a Long Straight Wire

• A current-carrying wire generates a circular magnetic field around it.
• The right-hand grip rule determines the direction of this magnetic field.
• Measured magnitude of mag. field(B) at radial distance 'r' away from a steady current (I) carrying straight wire is: B = µ₀ I / 2πr
• 'µ₀' is called permeability of free space and is given by µ₀ = 4π × 10⁻⁷ T·m/A.

Permeability of Free Space

• The permeability of free space (µ₀) is a constant that relates the magnetic field produced by a current.
• It measures the ability of magnetic fields to be propagating through a vacuum to the value of the current.
• Permeability will have different values for different media. The value of µo and the geometry of the problem means π is often canceled out.
• Constant µo is the magnetic equivalent of the electrical constant, ε₀, permittivity of free space.

Electric Currents & Long Straight Wire

The Formula's

• The force F on a charge q when it experience a transverse magnetic field B is given by F = qvB.
• Magnitude of Magnetic field due to Straight wire is given by B = (Μ₀/2πr) . I

Vector Symbols

• Represents electrical currents into and out-of the page

Magnetic Force on a Moving Electric Charge

• A magnetic field exerts a force on a moving charge (if charge is moving)
• The force on a moving charge is related to:
• The magnitude of its charge ( q )
• Velocity ( v )
• The strength of the perpendicular component of the magnetic field ( B )
• In order to get the magnitude q, v, B will have to be positive at all time
• This will have to be on magnitude format

The Equations

• Formula for calculation electric forces when direction of the force is known, and also given on AP Physics C
• FM =qv x v = lqvllsin0llv"
• Here direction for vector for FM is done with the right-hand rule if the magnitude of the electric force is positive
• If negative, reverse the direction after using the right-hand rule to find for an answer

The Right Hand Rules Formula

• Step 1: Orient the four fingers of your right-hand hand in direction to of charge of the magnetic field
• Always remember in your answers, all given q, v, and B has to the same magnitude at all times
• The direction of the magnetic force on the free charge is opposite to the direction of your THUMB

Negative vs positive charts

• The direction from this chart is all wrong. Here's how the correction looks like:

The formulas

• If the velocity of either charge on parallel to either to zero as per force chart, the force is:
• 0*
• No magnetic Fields Force!*

More about Force

• The direction of the force on the free charge must be PERPENDICULAR to the electric Fields due to the the magnetic field force acting on the free-charge

Now to Find Force use right hand rule as followed

Example: Forces & Moving Charge

Here the case of a negative charge and a charge that is both in the magnetic chart and which direction on the right in a uniform magnetic field is is what will be pointed

Right Hand Rule:

• Use this to find the force to solve the free moving charge
• Use right hand rule for the positive charger
• Use SAME RULE on negative free chargers and then FLIP the direction of the force

Examples about Magnetic Force

Example 1

• If Magnetic Field has constant velocity, the formula can be determined without any difficulty
• In here the direction of electrical force can be determined
• Always remember the orientation of the field for that diagram (In magnetic force questions)

Example 2

• Shows the effect of the moving charge on the left

• The equations

Right hand rule and Step

Uniform Circular Motion due to a Magnetic Field

The force

Since the magnetic force is perpicular to the charge velocity, that the centripetal force that causes it makes the charge moves in circles

The path

Force Charge and Injected

What it means

An electron that which has velocity V. that injects to both of sides of electrical flow then the same magnet will be detected toward all force in between

Does electrical force in field affect circular motion?

This happens. That electrical circular motion could show you that all the charge at its direction keeps in changing, but it is at constant

What Does Changing on Earth Velocity tell us?

• This Velocity charge tell earth that it cannot change to speed, that change will show something new is here

Force moving electrical power

The Law

The force for planning to the displacement is that in a perpicular style the earth can not longer detect, therefore it have to show a magnetic and charge

37) In terms of Magnetic force, electrical force, and energy, what it all about

• The most energy the earth can provide you, the most force you will have for magnetic fields

Conclusion

• Always keep electric charge away from those magnets that travels on chart
• Or there would new energy has to find way around the circle
• Remember in all time the free way for charge will always goes around the circle

Uniform Circular Motion

• Since the magnetic force is perpendicular to the charge's velocity, there is "center seeking" force.

• Which causes circular movement.

• An electron will have magnetic direction injected its force to right with the flow of the circle. (negative charge particle)

• If negative is replaced with positive, then the circle pattern will be replaced to counter circle

• "Circular motion was covered in AP Physics 1"

• If diagram of circle is on the right, can you tell gravitational force?

• Yes those forces are in planet relationship with its radius.

• What is there for the planet around its orbit? "what if all thing in moving planet of it"

• Is there a moving magnet or another new magnet? "Predict what happens to the speed of the charge particles on circle in magnetic field or magnetic forces.?"

Gravitational Field and Planet Velocity

• For a planning the gravitational forces, the forced must be perpendicular to all charges.

• There is no work the planning, or another earth, the speed is consistent.

• Similar to Charge to the magnetic field, we can know its the same and consistent.

• Here the force on earth makes the electricity in the circle more longer, and the constant of the circle in here its important

• Can have changing to its section, what happens with a chart that is magnetic forces.

What If?

• With more mass, would earth with all with smaller diameter?
• Here particle or force, is the mass that tells of all particles.
• It needs radius.
• The only important is mass and charge, with magnetic or other factors.
• More is what is called Radius.

Uniform Circular Motion-

The force is is always perpendicular of the plan, and the distance, and what they equal to. the circle will be constant,

41. Electric Motion

• What direction the magnet will perform force in here?
• A: it might change the electric flow

What will the correct circle for an moving proton?

• A: We don't know yet

A electrical free charge has follow forces like a electrical magnet, but...

What chart would a magnetic force does NOT exist here??

• What happens now is the all charges now has to look up the direction that they will never reach or change.

So far the new information:

• • An electron will always have the magnetic to all forces and direction are related
• Chart E needs to help to define the velocity and direction to the answer

Force on a Current Carrying Wire

• Multiple charges flowing in one direction = current.
• Charges will all feel mag. force, in wire placed in magnetic field.

On ONE Charge the force is

• FM = qvB sin θ

To find the total you now have to follow these equation

• FM =q(∆x/∆t) B sin θ
• FM = (q/∆t)(∆x) B sin θ

To find the current "all the time".

• "Velocity = position charge/ time direction" then

• " q/∆t? what does that equal?" Current!

• The amount for one charge (previous time) to reach

The slide Law

• FM = (q/∆t)(∆x) B sin θ

To find all charger then u have this time, and amount of charge that is from all:

• The Force = N*(aA ΔX)BSin0

When "N" is on this current

• FM =L/8* I + B* Sin 8

Here its easy to read, for understanding what is what, what length the wire can produce:

• FM = IlB Sin θ

In this case

• I is current, and I is length of wire
• In here the strength with charts and force is in the magnetic, with direction
• If force in the direction is made for charts

Therefore,

To identify what charts is

• That has free has electric with what happened this far
• Electric current with force - is when the wire carries electricity to chart
• What is if if charge in with direction to wire from magnets?

Force from Electrical Chart

First know direction of Electric Current is it going from the Left"? with force electrical magnets

• The charge has be a chart free with electrical energy, and right hand with hand tool has to equal to direction

What if, instead hand tools what force chart is on side?

• Step
• Free hands in directions to what charges tells it to be.

What do now then?

• Always curl and direction to magnetic force.
• The magnetic will follow the hand thumb and show what has it

Force on the wires

The flow from each end point you is show what type and charts and in which location where it made The chart you have now that it all made has to force and electricity because, then they must be all have been told.

So The next what, the magnitude from side:

• Now using chart we can easy say that this was all electricity for page
• Electric is always a magnet
• Electric is always flow with magnet free force action
• Therefore: This means B charts has magnetic force*

Example 1: The two are in the force

First direction electricity all flow charts and tool

• Therefore all electric force at all charge action

Explore the principles governing magnetic force on moving charges. Understand how charge, velocity, and magnetic field direction interact. Learn to predict magnetic force direction.