Electromagnetism II PHY1032S Handout PDF

Electromagnetism II - PHY1032S Dr Mawande Lushozi Mawande Lushozi Magnetism Consultation Email: [email protected] Room: Room 4.11 Physics Department Hours: Wed & Thur 12:30-13:30 Lectures: 15 Lectures... Then God Said,...

Electromagnetism II - PHY1032S Dr Mawande Lushozi Mawande Lushozi Magnetism Consultation Email: [email protected] Room: Room 4.11 Physics Department Hours: Wed & Thur 12:30-13:30 Lectures: 15 Lectures... Then God Said, 1 ∇⋅E= 𝜌 𝜖0 ∇⋅B=0 𝜕B ∇×E=− 𝜕𝑡 𝜕E ∇ × B = 𝜇0 J + 𝜇0 𝜖0 𝜕𝑡 AND THERE WAS LIGHT. Mawande Lushozi Magnetism How Hard Should You Be Working For every 1hr lecture, you need to put in 1hr of self study Problem sets and tutorials vital for passing course Work consistently throughout the course, cramming won’t work!!! You must use the textbook Mawande Lushozi Magnetism PHY1032S — Electricity & Magnetism – Part II Magnetism Magnetism, magnets and the magnetic field Magnetic force Torque and the electric Motor Force on moving point charges Electromagnetic Induction Induction and motional emf Generator Transformer PHY1032S — Electricity & Magnetism – Part II AC Circuits Alternating current Household electricity and power grid Resistors, inductors and capacitors RLC circuits Electromagnetic Waves Maxwell’s equations Electromagnetic waves PHY1032S — Is it going to be difficult? Spatial thinking Magnetism is inherently 3-dimensional Unlike electro-statics, which can be reduced to 2 dimensions Abstraction The concepts we discuss here may seem abstract. But they can explain many effects in the world around us. Compartmentalization of knowledge All areas of physics (and mathematics) are interlinked I will draw on previous knowledge: electricity, oscillations & waves, linear motion, angular motion, vectors... PHY1032S — Weekly Problem Sets WPS will be available as PDF on Amathuba. Solve them with pen and paper Submission of the problem set: Friday 10H00. PHY1032S — Magnetism Magnetism PHY1032S — Magnetism Permanent Magnets Let’s take a look at the force between Permanent Magnets attractive or repulsive long range force → magnetic field always two poles: North and South Chapter 11 | Magnetism when split, two smaller magnets remain Is this the electric force? No no force on charged objects at rest magnets not electrically charged New concepts: magnetic force and magnetic field PHY1032S — Magnetism Magnetic Monopoles? What does the theory say? no counter-arguments: all equations could N be formulated with magnetic monopoles some attractive features magnetic monopoles would be nice symmetry to electric charge interesting predictions from quantum mechanics: quantized charge... Is there experimental evidence? S Nothing, so far. Searches are ongoing... Discovery would be a sensation! Until magnetic monopoles have been discovered, we will assume that they do not exist. PHY1032S — Magnetism Magnetic Materials Other materials lose their magnetisation when the field is removed → Magnetic Materials “soft” magnetic materials always attracted by permanent magnets induced magnetic field parallel to external field “amplification” of magnetic field become magnetic in presence of another magnet no or little magnetization after external field is removed PHY1032S — Magnetism Electromagnets Solenoids can generate magnetic fields higher B field than straight wires, loops high currents required even for moderate fields Soft iron yoke can amplify the magnetic field: magnetization of iron core aligned with field from coil amplification of net magnetic field PHY1032S — Magnetism Towards the Magnetic Field Reminder: definition of the electric field E ~ =F ~ /q Not applicable — no magnetic monopoles! Alternative approach: An electric dipole in an electric field torque aligns dipole with E-field points in direction of field Idea: map out magnetic field lines using a magnetic dipole.  PHY1032S — Magnetism Field of a Bar Magnet Map out the field using many little compasses. Each compass points in the direction of the magnetic field. Magnetic field lines can be constructing from connecting the needles. PHY1032S — Magnetism Field of a Bar Magnet Comparison: the field of an electric dipole PHY1032S — Magnetism Field between two Magnets PHY1032S — Magnetism Magnetic Field of a Wire The magnetic field around a straight, long and current-carrying wire points in circles around the wire. PHY1032S — Magnetism Right Fist Rule But what is the magnitude of the magnetic field? PHY1032S — Magnetism Drawing Vectors and Currents in 3D Dots mark vectors/currents/... out of the page think of the tip of an arrow flying towards you Crosses mark vectors/currents/... into the page think of the feathers of an arrow flying away from you PHY1032S — Magnetism Magnetic Field of a Wire I PHY1032S — Magnetism Current Loop What is the magnetic field of a current loop? Direction of the magnetic field inside the loop (as shown in left picture) left — right — up — down — into the plane — out of plane  PHY1032S — Magnetism Magnetic Field of a Current Loop PHY1032S — Magnetism Magnetic Field of a Solenoid A solenoid is a series of current loops, stretched out over a length L PHY1032S — Magnetism Forces between Currents Direction of the magnetic force Field lines around a wire are known. Attraction or repulsion between (anti-)parallel currents? What is the orientiation of current, magnetic field, magnetic force? Magnitude of the magnetic force Force between wires can be measured What can we learn for the magnitude of the magnetic field? PHY1032S — Magnetism Magnetic Fields and Magnetic Force between Wires F ~ F ~ I I F ~ F ~ I I PHY1032S — Magnetism Forces between Currents Direction of magnetic force on current in wire: perpendicular to wire / current perpendicular to magnetic field PHY1032S — Magnetism Forces between Currents Magnitude of force on a wire: proportional to length L proportional to current in both wire I1 , I2 inversely proportional to wire distance r and some constants µ0 /2π µ0 LI1 I2 Fparallel wires = 2πr The constant µ0 is called (magnetic) permeability. It is fixed (by the definition of the ampere in SI units) as: µ0 = 4π × 10−7 N2 C−1 PHY1032S — Magnetism Worked Example: Magnetic force between two wires Two 2 m long wires with a mass of 100 g each are suspended by 0.800 m strings as shown in the figure. The wires carry equal currents in opposite directions, i.e. the current flowing on one wire returns a = 0.800 m on the other one. The distance between the wires is measured as 7 mm. What is the magnetic force between the wires? d = 7 mm What is the current in the wires? PHY1032S — Magnetism Magnetic Field around a Wire We use this to define the magnitude of the magnetic field |B| ~ due to wire 1 at the position of wire 2: Fparallel wires µ 0 I1 |B| ~ = = LI2 2πr The magnetic field lines are circles around the wire; the direction is given by the right fist rule. The unit of the magnetic field is 1 tesla = 1 T = 1 N/Am The magnetic permeability µ0 has the units µ0 = 4π × 10−7 T m A−1  PHY1032S — Magnetism Magnetic Force on a Straight, Current-Carrying Wire The magnetic force is perpendicular to the current in the wire perpendicular to the magnetic field The direction of the magnetic force can be determined with the right hand rule thumb ↔ current index finger ↔ magnetic field middle finger ↔ magnetic force The magnitude of the magnetic force is: FB = ILB sin α F ~ B = I~L × B ~ PHY1032S — Magnetism Cross Product ~c ~c = ~a × ~b ~b |~c | = |~a | |~b| sin α ~a  PHY1032S — Magnetism Magnetic Field of a Current Loop PHY1032S — Magnetism Current Loops and Thin Coils B field at the center of a current loop µ0 I B= 2R Combining N current loops into a thin coil creates a higher B-field µ0 NI B= 2R A coil is “thin” if the cross section of all wires is much smaller than the loop area.  PHY1032S — Magnetism Magnetic Field of a Solenoid A solenoid is a series of current loops, stretched out over a length L PHY1032S — Magnetism Magnetic Field inside a Solenoid The magnetic field inside an infinitely long solenoid is uniform with magnitude: N B = µ0 I L PHY1032S — Magnetism Example: Solenoid We have a tightly wound solenoid with a length of 20 cm and a diameter of 1 cm. The solenoid is made from a thin copper wire with a diameter of 0.200 mm. What current is required to create a magnetic field of 0.0100 T inside the solenoid? What is the power dissipated by wire due to this current? Note: the resistivity of copper is ρ = 1.70 × 10−8 Ω m PHY1032S — Magnetism Magentic Field of the Earth The earth is a large magnet the north pole of a magnet points north, towards the south magnetic pole → compass the south magnetic pole is near the north geographic pole (and vice versa) deviation: difference between magnetic and geographic north magnetic field lines enter the earth’s surface at an angle The magnitude of the earth’s magnetic field depends on the location. On the surface of the earth, it ranges around B ≈ 25 µT − 65 µT PHY1032S — Magnetism Map of Magnetic Declination US/UK World Magnetic Model -- Epoch 2010.0 Main Field Declination (D) 180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180° 70°N -40 70°N -10 -10 10 0 20 -3 10 0 60°N 60°N -10 0 -2 20 45°N 45°N 0 30°N 30°N 0 15°N 10 15°N 0 -1 0° 0 0° -10 -2 10 0 -10 15°S 15°S 0 -20 30°S 30°S 10 -30 20 20 45°S 45°S 30 30 0 -6 10 40 40 20 60°S -7 0 60°S 50 50 -90 60 0 j k 70 -8 -100 60 100 9 80 30 20 13 0 70 10 -1 -1 -1 0 110 -30 -50 -20 -40 8 70°S 0 70°S 180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180° Main field declination (D) Map developed by NOAA/NGDC & CIRES Contour interval: 2 degrees, red contours positive (east); blue negative (west); green (agonic) zero line. http://ngdc.noaa.gov/geomag/WMM/ Mercator Projection. Map reviewed by NGA/BGS j : Position of dip poles Published January 2010 PHY1032S — Magnetism Measuring Magnetic Fields Compass used for millenia shows direction, but not magnitude Electronic Compass / Smartphone measures direction and magnitude how does it work? → Hall effect PHY1032S — Magnetism Ferromagnets Permanent Magnets create a static magnetic field Magnetic Materials strongly attracted by magnets Magnets and magnetic materials are usually made of iron, nickel, cobalt → Ferromagnetism PHY1032S — Magnetism Ferromagnetism atoms are miniature magnets unordered magnetic moments → no net magnetisation aligned magnetic moments in ferromagnets→ strong magnetic field  PHY1032S — Magnetism Induced Magnetic Moments r 22 | Magnetism 426 Chapter 11 | Magnetism 22.8 (a) An unmagnetized piece of iron (or other ferromagnetic material) has randomly oriented domains. (b) When magnetized by an ex Unmagnetized e domains show greater alignment, andiron some grow at the expense ofMagnetization inareexternal others. Individual atoms B-Field aligned within domains; each atom act magnet. regions (“domains”) of domains aligned with the B-field aligned magnetic moments grow ersely, a permanent magnet can be demagnetized by hard blows or by heating it in the absence of another magnet. sed thermal motion at higher temperature can disrupt and randomize the orientation and the size of the domains. The -defined temperature Figuredifferent 11.6 North domains for ferromagnetic and south arein pairs. Attempts materials, poles always occur which is called to separate other thein more them result Curie domains poles. If we shrink temperature, pairs of continue toabove which split the magnet, we they cannot be randomly aligned will eventually get down to an iron atom with a north pole and a south pole—these, too, cannot be separated. etized. The Curie temperature for iron is 1043 K (770ºC) , which is well above room temperature. There are several material becomes magnetized The fact that magnetic poles always occur in pairs of north and south is true from the very large scale—for example, sunspots nts and alloys always that have occur inCurie temperatures pairs that are north and southmuch lower magnetic than poles—all the room way downtemperature to the very smalland scale.are ferromagnetic Magnetic atoms have only below tho ratures. both a north pole and a south pole, as do many types of subatomic particles, such as electrons, protons, and neutrons. romagnets Making Connections: Take-Home Experiment—Refrigerator Magnets We know that like magnetic poles repel and unlike poles attract. See if you can show this for two refrigerator magnets. Will n the 19th PHY1032S century, it—was the magnets discovered Magnetism stick that if you turn them electrical over? Why do theycurrents cause stick to the door magnetic anyway? What caneffects. The you say about thefirst significant observation w magnetic Danish scientistproperties Hans ofChristian the door next Oersted (1777–1851), to the magnet? who found Do refrigerator magnets stick to that a plastic metal or compassspoons?needle wasto deflected Do they stick all types by a current- Creating a Permanent Magnet of metal? ng wire. This was the first significant evidence that the movement of charges had any connection with magnets. omagnetism is the use of electric current to make magnets. These temporarily induced magnets are called 11.2 Ferromagnets omagnets. Electromagnets and Electromagnets are employed for everything from a wrecking yard crane that lifts scrapped cars to contro am of a 90-km-circumference Ferromagnets particle accelerator to the magnets in medical imaging machines (See Figure 22.9). Hard Magnetic Material Only certain materials, such as iron, cobalt, nickel, and gadolinium, exhibit strong magnetic effects. Such materials are called Soft ferromagnetic, after the Latin word for iron, ferrum. A group of materials made from the alloys of the rare earth elements are Stable location Move easily also used as strong and permanent magnets; a popular one is neodymium. Other materials exhibit weak magnetic effects, which Domain Walls 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. Figure 11.7 An unmagnetized piece of iron is placed between two magnets, heated, and then cooled, or simply tapped when cold. The iron becomes a permanent magnet with the poles aligned as shown: its south pole is adjacent to the north pole of the original magnet, and its north pole is adjacent to the south pole of the original magnet. Note that there are attractive forces between the magnets. Create a permanent magnet: When a magnet is brought near a previously unmagnetized ferromagnetic material, it causes local magnetization of the material with unlike poles closest, as in Figure 11.7. (This results in the attraction of the previously unmagnetized material to the magnet.) take hard magnetic material What happens on a microscopic scale is illustrated in Figure 11.8. The regions within the material called domains act like small bar magnets. Within domains, the poles of individual atoms are aligned. Each atom acts like a tiny bar magnet. Domains are heat (or tap) → ease moving of domain walls → “soften” material 22.9 Instrument small for magnetic goes into this “tunnel” and randomly may growontothe resonance oriented inimaging gurney. millimeter size, (MRI).ferromagnetic an unmagnetized (credit: aligningBill The deviceobject. McChesney, themselves as shownFlickr) uses Ina response superconducting cylindrical field. T to an external coil the magnetic field, in Figure 11.8(b). This induced magnetization can be made fordomains the main magnetic apply external field → magnetize permanent if the material is heated and then cooled, or simply tapped in the presence of other magnets. e 22.10 shows that the response of iron filings to a current-carrying coil and to a permanent bar magnet. The patterns cool down/stop r. In fact, electromagnets tapping/harden and ferromagnets have the samematerial → magnetization basic characteristics—for remains example, they have north and so that cannot be separated and for which like poles repel and unlike poles attract. This OpenStax book is available for free at https://legacy.cnx.org/content/col12128/1.2  ombining a ferromagnet PHY1032S — Magnetism with an electromagnet can produce particularly strong ma trong magnetic effects are needed, such as lifting scrap metal, or in particle accele Solenoid erromagnetic with soft-iron materials. corestrong the magnets can be made are impose Limits to how t sufficiently high current), and so superconducting magnets may be employed. Th roperties are destroyed by too great a magnetic field. Soft magnetic materials can be used to guide and enhance magnetic fields. Soft iron core is inserted into solenoid Magnetic field of solenoid aligns domains in core Soft iron core becomes magnetized Induced magnetization can be much stronger than solenoid’s field. gure 11.11 An electromagnet with a ferromagnetic core can produce very strong magnetic effe agnet, the poles of which are aligned with the electromagnet. igure 11.12 shows a few uses of combinations of electromagnets and ferromagne memory devices, because the orientation of the magnetic fields of small domains c nformation storage PHY1032S on videotapes and computer hard drives are among the most c — Magnetism ur digital world. Torque on a Current Loop The forces on opposite sides are opposite and equal, but for the top and bottom sides, they do not act along the same line. We therefore have no net force Fnet = 0 torque on the loop τ = IAB sin θ (see tutorial on how to calculate this) PHY1032S — Magnetism Worked Example: Triangular Current Loop A equilateral triangle is placed in a uniform magnetic field, with the plane of the triangle parallel to the magnetic field: B ~ I What are the net force and the torque on the triangle? PHY1032S — Magnetism Magnetic Dipole Moment The equation for the torque on a current loop τ = IA B sin θ only depends on the product of loop area A and current I. We introduce the magnetic dipole moment m of a current loop as m ~ = IA ~ The length of the vector A ~ is the area of the loop, and the direction is perpendicular to the plane of the loop. Remember that the fields of loops, solenoids and bar magnets look similar. The concept of a magnetic dipole moment therefore even applies to bar magnets, solenoids etc.  PHY1032S — Magnetism Torque on a Magnetic Dipole With the magnetic dipole moment m ~ we can write the torque on a dipole: ~τ = I A ~ ×B~ =m ~ ×B ~ m| |B| |~τ | = |~ ~ sin θ The torque on the dipole will align the magnetic dipole moment with the magnetic field. Chapter 22 | Magnetism 875 For sin Ȇ = 1 , the maximum torque is Ȓ max = /*"#. (22.22) Entering known values yields Ȓ max = (100)(15.0 A)130.100 m 246(2.00 T) (22.23) = 30.0 N ⋅ m. Discussion This torque is large enough to be useful in a motor. PHY1032S — Magnetism The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at Ȇ = 0. The Electric Motor torque then reverses its direction once the coil rotates past Ȇ = 0. (See Figure 22.35(d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at Ȇ = 0. To get the coil to continue rotating in the same direction, we can reverse the current as it passes through Ȇ = 0 with automatic switches called brushes. (See Figure 22.36.) Ȇ =moment Torque turns loop → magnetic dipole aligned 0 , the brushes Figure 22.36 (a) As the angular momentum of the coil carries it through with magnetic field reverse the current to keep the torque clockwise. (b) The coil will rotate continuously in the clockwise direction, with the current reversing each half revolution to maintain the clockwise torque. Reverse direction of current → loop turns to opposite direction Meters, such as those in analog fuel gauges on a car, are another common application of magnetic torque on a current-carrying Commutator: inverts current direction every half turn loop. Figure 22.37 shows that a meter is very similar in construction to a motor. The meter in the figure has its magnets shaped to limit the effect of Ȇ by making # perpendicular to the loop over a large angular range. Thus the torque is proportional to * PHY1032S — Magnetism Electric Motors in Practice τ = m B sin θ Magnetic dipole moment m and field B Largest torque for high dipole moment of rotor, strong magnetic field of stator Loops with many turns Soft iron core to amplify fields Angle θ between dipole moment and field Highest torque for sin θ ≈ 1, i.e. θ ≈ 90° Dead points (no torque) for sin θ = 0, i.e. θ = 0°, 180° Use rotor with many poles PHY1032S — Magnetism Experiment: Electron Beam in Magnetic Fields  PHY1032S — Magnetism Magnetic Force on a Charged Particle A magnetic field deflects charged If the particle is not moving particles moving perpendicular to it. perpendicular to the field, it is deflected less. PHY1032S — Magnetism Magnetic Force on a Charged Particle The magnetic field does not deflect a A particle at rest does not experience particle moving parallel to it. a magnetic force. PHY1032S — Magnetism Force on a Point Charge A particle with charge q and velocity ~v is moving in a magnetic field B. ~ What is the force on the particle? We start from the force on a current: F ~ = I~L × B. ~ Current I, distance travelled ~L, charge q and velocity ~v are linked: q  q  I= L = ~v ∆t ~ ⇒ IL = ~ (~v ∆t) = q~v ∆t ∆t We can then write the magnetic force on this point charge: F ~ = q~v × B ~ PHY1032S — Magnetism Point Charge in Uniform Magnetic Field A point charge q is moving in a uniform magnetic field. The angle between the velocity of the particle ~v and the magnetic field B ~ is α. ~v q B ~ What is the trajectory of the point charge?  PHY1032S — Magnetism Force on the Point Charge in Uniform Magnetic Field The force on the point charge F ~ = q~v × B ~ is perpendicular to the velocity and the magnetic field. The component of the force parallel to the magnetic field, Fk , is therefore 0. The motion in this ~v direction is uniform: F ~ B ~ ak = Fk /m = 0 vk = const xk = vk t + const PHY1032S — Magnetism Reminder: Circular Motion What is the force required to keep an object on a circular path? the acceleration, and hence the force, is perpendicular to the velocity F ~ ⊥ ~v the magnitude of the force is 2 mv 2 Fcent = mr ω = r PHY1032S — Magnetism Charged Particle in Magnetic Field The magnetic force is always perpendicular to the velocity F ~ = q~v × B ~ In mechanics, we fixed the radius and found the string tension (force) to keep an object in circular motion. For a charged object moving in a magnetic field we know the magnetic force and use it to determine the radius of its circular path. PHY1032S — Magnetism Example: Motion Perpendicular to Magnetic Field We equate magnetic and centripetal force: Fmag = Fcent mv⊥2 qv⊥ B = r And we solve for the radius of the circular path: mv⊥ r= qB PHY1032S — Magnetism Motion of a Charged Particle in a Magnetic Field Superposition of two components: parallel to magnetic field no force in direction of B ~ uniform motion with velocity, momentum vk = v cos α pk = p cos α perpendicular to magnetic field force perpendicular to velocity circular motion with velocity and radius v⊥ = v sin α p⊥ = p sin α mv sin α mv⊥ p⊥ r= = = qB qB qB The superposition results in a helix path for the moving charged particle. The particle path curls around the field lines PHY1032S — Magnetism Mass Spectrometer The motion of a point charge is determined by its charge q, the mass m and the velocity ~v. In particle, nuclear or atomic physics, the charge is usually a small multiple of the elementary charge. Often, it is simply ±e. We are then left with 2 unknowns — m and ~v — and we can use two of the three following quantities to fix them: ~v ~p = m~v Ekin = 12 mv 2 We have just seen how we can use a magnetic field to measure the momentum We often use an electric field to accelerate the particles to a certain kinetic energy. A combindation of electric and magnetic fields can be used as a velocity selector. PHY1032S — Magnetism Mass Spectrometer PHY1032S — Magnetism Example: Mass Spectrometer Ions are accelerated by an electric potential difference of 1500 V and deflected in a magnetic field of 10 mT. The bending radius is measured to be 2.95 cm. What is the mass of the ions? Acceleration: 1 2 Ekin − Ekin,initial = −∆U ⇔ mv − 0 = −q∆V 2 Analyser: mv 2 rqB Fmag = Fcent ⇔ qvB = ⇒ v= r m We combine the two: 1 rqB 2 r 2 q 2 B 2   m = = −q∆V 2 m 2m m r 2B 2 0.100 kg 5.98 × 1025 u ⇒ = = = q 2∆V e e  Solving this for the Hall emf yields PHY1032S — Magnetism where is the Hall effect voltage across a conductor of width through which charges move at a speed. Hall Effect The magnetic force also acts on electrons inside a conductor in a magnetic field. F ~ = q~v × B ~ B field deflects electrons to one side of conductor excess charges create electric field in conductor and potential difference V between sides Figure 11.28 The Hall emf produces an electric force that balances the magnetic force on the moving charges. The magnetic force pr charge separation, which builds up until it is balanced by the electric force, an equilibrium that is quickly reached. Steady state: electric andOnemagnetic forces of the most common cancel uses of the Hall effect is in the measurement of magnetic field strength. Such devices, c probes, can be made very small, allowing fine position mapping. Hall probes can also be made very accurate, usually qV accomplished by careful calibration. Another application of the Hall effect is to measure fluid flow in any fluid that has Felecshown. = qE Fmagnot = qvB charges (most do). (See Figure 11.29.) A magnetic field applied perpendicular to the flow direction produces a Hall e =the sign of =depends Note that of the Hall emf is l on the sign of the charges, but only on the directions of , where is the pipe diameter, so that the average velocity and. The m can be determined from the other factors are known. We can measure a potential difference between the sides: VHall = vlB PHY1032S — Magnetism Hall Sensor The Hall effect creates a potential difference – the Hall voltage – between the sides of a conductor that proportional to the magnetic field: VHall = vlB This can be used to measure a magnetic field. Semiconductor Hall sensors can be produced very cheaply and are used e.g. in many smartphones. PHY1032S — Electromagnetic Induction Electromagnetic Induction A change of the magnetic field near a solenoid induces a current. PHY1032S — Electromagnetic Induction Motional EMF We move a conducting bar through a magnetic field. What happens to the charge carriers in the bar? Ions experience a force, but are fixed in position → no effect Electrons experience a force in the opposite diretion and follow it → accumulate at end of bar. Let’s assume for the argument that we have positive free charge carriers... PHY1032S — Electromagnetic Induction Motional EMF Let’s assume for the argument that we have positive free charge carriers. (positive) charge carriers accumulate at end of bar (negative) net charge remains at opposite end the separation of charges creates an electric field In equilibrium the electric and magnetic forces cancel: ∆V FB = FE ⇒ qvB = qE = q L We see a potential difference between the ends of the bar: ∆V = L E = v L B PHY1032S — Electromagnetic Induction Induced Current in a Circuit keep 3 sides of loop fixed → no induced voltage move 4th side of loop → induced voltage result: current in loop Electromagnetic Induction PHY1032S — Electromagnetic Induction Magnetic Force on Induced Current Magnetic force on current points to the left slows down rod (moving to the right) Pulling force required to maintain speed PHY1032S — Electromagnetic Induction Power and Energy pulling force does work on moving rod work is converted to electric energy electric energy is dissipated as heat (resistance of wires) Convert mechanical to electric energy PHY1032S — Electromagnetic Induction Induced Current in a Loop a a Move two sides in opposite directions b b cos θ Only motion perpendicular to B-field counts Rotating the loop is b b cos θ θ a convenient way to achieve this PHY1032S — Electromagnetic Induction Induced Currents? v v v v v v v PHY1032S — Electromagnetic Induction Magnetic Flux The “number of field lines” is not a precise quantity. We therefore introduce the a magnetic flux: Φ = AB cos θ = A ~ ·B ~ b cos θ Where A is the area of the loop — for a rectangular loop A = ab, but the equation holds for any shape B is the magnetic field strength b cos θ b θ θ is the angle between the magnetic field and the axis of the loop The unit of the magnetic flux is 1 Tm2 = 1 Weber = 1 Wb PHY1032S — Electromagnetic Induction Lenz’s Law There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. Nature dislikes a change in magnetic flux. PHY1032S — Electromagnetic Induction Eddy Currents PHY1032S — Electromagnetic Induction Faraday’s Law An emf E is induced in a conducting loop if the magnetic flux through the loop changes. For a change of the magnetic flux ∆Φ in a time interval ∆t, the induced emf is: ∆Φ E =− ∆t The direction of the emf can also be determined by Lenz’s law. PHY1032S — Electromagnetic Induction Analogy: Simple Harmonic Motion We know the Special case: magnetic flux y(t and= want 0) = Ato calculate the motional emf: Φ(t) = N A ~ ·B ~ = NAB cos(ωt) ∆Φ E(t) = − =? ∆t You know another system with sinusoidal time dependence: the simple harmonic oscillator: x(t) = A cos(2πft) v(t) = −(2πf ) A sin(2πft) a(t) = −(2πf )2 A cos(2πft) Can we use this knowledge? PHY1032S — Electromagnetic Induction Analogy: Simple Harmonic Motion We can use the analogy to the SHM to solve our problem: ∆x ∆A cos(2πft) ∆ cos(2πft) = =A = −(2πf ) A sin(2πft) ∆t ∆t ∆t We see (with ω = 2πf ): ∆ cos(ωt) ∆x→0 −−−−→ −ω sin(ωt) ∆t And we can put this into our equation for the induced emf: ∆Φ ∆NAB cos(ωt) ∆ cos(ωt) E =− = = NAB = −ω NAB sin(ωt) ∆t ∆t ∆t PHY1032S — Electromagnetic Induction Summary: Simple Harmonic Motion We can compare SHM and our equations for magnetic flux and induced emf: x(t) = A cos(2πft) Φ(t) = NAB cos(ωt) ∆ cos ωt v(t) = −(2πf ) A sin(2πft) E(t) = −ωNAB ∆t a(t) = −(2πf )2 A cos(2πft) We remember: ∆ cos(ωt) ∆t→0 −−−−→ −ω sin(ωt) ∆t ∆ sin(ωt) ∆t→0 −−−−→ ω cos(ωt) ∆t PHY1032S — Electromagnetic Induction Generator A generator is a device that converts mechanical energy into electrical energy that can e.g. be used to drive a current through a circuit. The output of the generator is an AC (alternating current) voltage: V (t) = V0 sin ωt = NABω sin ωt PHY1032S — Electromagnetic Induction Generators PHY1032S — Electromagnetic Induction Generators  Ȓ max = /*"#. (22.22) PHY1032S — Electromagnetic Induction Entering known values yields Ȓ max = (100)(15.0 A)130.100 m 246(2.00 T) (22.23) Back Emf = 30.0 N ⋅ m. Discussion This torque is large enough to be useful in a motor. The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at Ȇ = 0. The torque then reverses its direction once the coil rotates past Ȇ = 0. (See Figure 22.35(d).) This means that, unless we do Generators and electric motors are very similar: something, the coil will oscillate back and forth about equilibrium at Ȇ = 0. To get the coil to continue rotating in the same direction, we can reverse the current as it passes through Ȇ = 0 with automatic switches called brushes. (See Figure 22.36.) Motors are also generating an induced voltage (called Ȇ=0 back emf): Figure 22.36 (a) As the angular momentum of the coil carries it through , the brushes reverse the current to keep the torque clockwise. (b) The coil will rotate continuously in the clockwise direction, with the current reversing each half revolution to maintain the clockwise torque. torque accelerates rotor Meters, such as those in analog fuel gauges on a car, are another common application of magnetic torque on a current-carrying loop. Figure 22.37 shows that a meter is very similar in construction to a motor. The meter in the figure has its magnets shaped to limit the effect of Ȇ by making # perpendicular to the loop over a large angular range. Thus the torque is proportional to * increasing ω → increasing back emf and not Ȇ. A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection * proportional to. If an exact proportionality cannot be achieved, the gauge reading can be calibrated. To produce a galvanometer for use in analog voltmeters and ammeters that have a low resistance and respond to small currents, we use a induced back emf reduces current, and torque large loop area " , high magnetic field # , and low-resistance coils. equilibrium when back emf equals supply voltage PHY1032S — Electromagnetic Induction Changing Magnetic Field A change in the effective area A cos θ of the loop is not the only way to change the flux, we can also change the magnetic field: Φ = AB cos θ According to Faraday’s Law, we should also get an induced current if the magnetic field in the solenoid changes: PHY1032S — Electromagnetic Induction Worked Example: Changing Magnetic Field A solenoid with 500 turns, length L = 20 cm and rS = 10 mm carries a sinusoidal current I(t) = I0 cos ωt = (0.0500 A) cos (2π(50 Hz)t) The solenoid is filled with a soft iron core increases the magnetic field by a factor 1000. An aluminium ring with radius rL = 50 mm, width w = 10 mm and thickness d = 1 mm surrounds the loop. What is the induced current in the loop? Note: the resistivity of aluminium is ρAl = 2.65 × 10−8 Ω m PHY1032S — AC Circuits AC Electricity PHY1032S — AC Circuits Generator A generator is a device that converts mechanical energy into electrical energy. The output of the generator is an AC (alternating current) voltage V (t) = V0 sin ωt PHY1032S — AC Circuits AC Voltage Source AC voltage source: provides time-dependent emf with periodically changing polarity. The emf is usually sinoidal: 2πt   V (t) = V0 cos ωt = V0 cos(2πft) = V0 cos T instantaneous, time dependent quantities V (t), I(t), P(t)... amplitudes or peak values: V0 , I0... f : frequency = oscillations per second (power grid: f = 50 Hz = 50 /s) ω: angular frequency ω = 2πf (in rad s−1 ) T : period = duration of one oscillation (T = 1/f = 25 ms) PHY1032S — AC Circuits Resistor: Voltage and Current V (t) V0 cos ωt I R V0 t Input voltage: V (t) = V0 cos(2πft) I(t) I0 Ohm’s Law is valid at all times: V (t) t I(t) = R The current is then The current through a resistor is V0 in phase with the voltage. I(t) = cos(ωt) = I0 cos(ωt) R PHY1032S — AC Circuits AC Power in Resistors V (t) We start from V0 V (t) = V0 cos(ωt) t V (t) V0 I(t) = = cos(ωt) = I0 cos(ωt) R R I(t) The instantaneous power P(t) is then I0 P(t) = I(t) V (t) = I0 V0 cos2 (ωt) t The peak power is P(t) Ppeak = I0 V0 Ppeak The average power is often more relevant: 1 t hPi = IV cos2 (2πft) = I0 V0 2 PHY1032S — AC Circuits RMS Current and Voltage If we define V0 I0 Vrms = √ and Irms = √ 2 2 then we can write I0 V0 I V 2 Vrms    P= = √0 √0 = Irms Vrms = 2 Irms R = 2 2 2 R The expressions for AC power are identical to those used for DC currents if rms currents and voltages are used. PHY1032S — AC Circuits Household Electricity The power grid runs at a frequency of f = 50 Hz, and households are supplied with a voltage of Vrms = 220 V. What is the instantaneous voltage? V (t) = (220 V) cos (50 s−1 )t

V (t) = (220 V) cos (314 s−1 )t
V (t) = (311 V) cos (50 s−1 )t
V (t) = (311 V) cos (314 s−1 )t What are peak and rms current in a 1200W heater? P 1200 W √ Irms = = = 5.5 A I0 = Irms 2 = 7.7 A Vrms 220 V What peak power consumed by the heater? √ 2 Ppeak = V0 I0 = Vrms Irms 2 = 2P = 2400 W PHY1032S — AC Circuits Changing Magnetic Field A change in the effective area A cos θ of the loop is not the only way to change the flux, we can also change the magnetic field: Φ = AB cos θ According to Faraday’s Law, we should also get an induced current if the magnetic field in the solenoid changes: PHY1032S — AC Circuits Transformer AC current creates oscillating B-field iron core “guides” magnetic flux to 2nd coil oscillating B-field induces voltage PHY1032S — AC Circuits Transformer The (change of) magnetic flux is equal for both coils: ∆Φ ∆Φ1 ∆Φ2 = = ∆t ∆t ∆t Using Faraday’s Law, we get the voltages for both coils: ∆Φ V1 = N1 ∆t ∆Φ V2 = N2 ∆t The change of magnetic flux cancels, and we get: V1 V2 V2 N2 = or = N1 N2 V1 N1 PHY1032S — AC Circuits Currents in the Transformer Let’s assume solenoids on both sides of the transformer with radius r , length L and N1 or N2 turns, respectively. µ0 N 1 I1 µ0 N 2 I2 Φ = AB cos α = (πr 2 ) cos(90°) = (πr 2 ) cos(90°) L L In this case, most quantities cancel and we find I2 N1 N 1 I1 = N 2 I2 ⇔ = I1 N2 We can link currents and voltages, using Φ = kN1 I1 = kN2 I2 ∆Φ V1 = N1 ∆t PHY1032S — AC Circuits Power and Transformers Energy is conserved → the power into a transformer equals the power out: P1 = P2 ⇔ V1 I1 = V2 I2 Currents are transformed inversely proportional to voltages: N1 V1 I2 = = N2 V2 I1 We can use a transformer to change a high voltage / low current to low voltage / high current. PHY1032S — AC Circuits Power Transmission Why are long-distance power lines run at high voltages? PHY1032S — AC Circuits Power Transmission ˜ The power dissipated (lost to heat) by a cable is 2 P = Irms R To reduce the loss, we can use transformers on both sides of a power line to decrease the current (and increase the voltage). PHY1032S — AC Circuits Household Electricity Three wires / contacts: line, hot, phase, active: carries current, oscillates with ≈ v = V (t) neutral: carries current, near potential of ground ground: direct connection to earth / ground near building, does not carry current unless a wiring error occurs  A thermal hazard occurs PHY1032S when — AC there is electrical overheating. A shock hazard occurs when electric Circuits erson. Both hazards have already been discussed. Here we will concentrate on systems and devices Electrical Safety ds. hematic for a simple AC circuit Figure 23.31 with no safety Schematic of afeatures. Thiscircuit simple AC is notwith howapower is distributed voltage source andina practice. single appliance represented by the resistan strial wiring requires the three-wire system, features in this circuit. shown schematically in Figure 23.32, which has t is the familiar circuit breaker (or fuse) to prevent thermal overload. Second, there is a protective such as a toaster or refrigerator. The case’s safety feature is that it prevents a person from touching nto electrical contact with the circuit, helping prevent shocks. 3. There are no safety Top: basic AC circuit mple AC circuit with a voltage source and a single appliance represented by the resistance without safety features Right: schematic of 3-wire circuit with safety features Figure 23.32 The three-wire system connects the neutral wire to the earth at the voltage source and user location, forcin supplying an alternative return path for the current through the earth. Also grounded to zero volts is the case of the appli protects against thermal overload and is in series on the active (live/hot) wire. Note that wire insulation colors vary with r check locally to determine which color codes are in use (and even if they were followed in the particular installation). r 23 | Electromagnetic ThereInduction, PHY1032S — ACthree are ACconnections Circuits Circuits, and Electrical to earth Technologies or ground (hereafter referred to as “earth/ground”) shown in Figu earth/ground connection is a low-resistance path directly to the earth. The two earth/ground connection Electrical Safety it to be at zero volts relative to the earth, giving the wire its name. This wire is therefore safe to touch e white, istomissing. The neutral wire is theand return path to foroperate the current to follow toFigure complete theshows circuit.aFur re (hereafter referred as “live/hot”) supplies voltage current the appliance. 23.33 mo al temversion of how connects the ground neutralthe connections wirethree-wire supply to the earth at system an the voltageissource alternative connected path through and user location, through the a three-prong forcing earth, a plugand it to be at zero volts good conductor, to an appliance. to complete the cir connection ath for the current through the earth.closest to the Also grounded power to zero volts source is the casecould be at the of the appliance. generating A circuit breaker orplant, fuse while the other is at the user’s d and is in series onground the activeis (live/hot) to thewire. caseNoteof that wireappliance, the insulation colors vary withthe through region and it is green essential to earth/ground wire, forcing the case, too, to be color codes are in use (and even if they were followed in the particular installation). s to earth or ground (hereafter referred to as “earth/ground”) shown in Figure 23.32. Recall that an a low-resistance path directly to the earth. The two earth/ground connections on the neutral wire force This to the earth, giving OpenStax the book This wire its name. is available for free atsafe wire is therefore http://cnx.org/content/col11406/1.9 to touch even if its insulation, usually al wire is the return path for the current to follow to complete the circuit. Furthermore, the two earth/ an alternative path through the earth, a good conductor, to complete the circuit. The earth/ground wer source could be at the generating plant, while the other is at the user’s location. The third earth/ appliance, through the green earth/ground wire, forcing the case, too, to be at zero volts. The live or for free at http://cnx.org/content/col11406/1.9 23.33 The standard three-prong plug can only be inserted in one way, to assure proper function of the three-wire system. e on insulation color-coding: Insulating plastic is color-coded to identify live/hot, neutral and ground wires but these co  PHY1032S — AC Circuits Electrical Safety e 23.34 Worn insulation allows the live/hot wire to come into direct contact with the metal case of this appliance. (a) The earth/ground con broken, the person is severely shocked. The appliance may operate normally in this situation. (b) With a proper earth/ground, the circuit b orcing repair of the appliance. PHY1032S — AC Circuits omagnetic induction causes a more subtle problem that is solved by grounding the case. The AC current in appliance Electrical Safety e an emf on the case. If grounded, the case voltage is kept near zero, but if the case is not grounded, a shock can oc ed in Figure 23.35. Current driven by the induced caseChapter emf is23 | Electromagnetic Induction, AC Circuits, and Electrical Techno called a leakage current, although current does not ssarily pass from the resistor to the case. penStax book is available for free at http://cnx.org/content/col11406/1.9 ound fault interrupter PHY1032S (GFI) — AC is a safety device found in updated kitchen and bathroom wiring that works based on Circuits romagnetic induction. GFIs compare the currents in the live/hot and neutral wires. When live/hot and neutral currents are not Electrical Safety — Ground Fault Interrupter (GFI) l, it is almost always because current in the neutral is less than in the live/hot wire. Then some of the current, again called a age current, is returning to the voltage source by a path other than through the neutral wire. It is assumed that this path ents a hazard, such as shown in Figure 23.36. GFIs are usually set to interrupt the circuit if the leakage current is greater 5 mA, the accepted maximum harmless shock. Even if the leakage current goes safely to earth/ground through an intact h/ground wire, the GFI will trip, forcing repair of the leakage. e 23.36 A ground fault interrupter (GFI) compares the currents in the live/hot and neutral wires and will trip if their difference exceeds a safe. The leakage current here follows a hazardous path that could have been prevented by an intact earth/ground wire. re 23.37 shows how a GFI works. If the currents in the live/hot and neutral wires are equal, then they induce equal and site emfs in the coil. If not, then the circuit breaker will trip. PHY1032S — AC Circuits RLC Circuits e 23.37 A GFI compares currents by using both to induce an emf in the same coil. If the currents are equal, they will induce equal but opposite PHY1032S — AC Circuits Resistor: Voltage and Current V (t) V0 cos ωt I R V0 t Input voltage: V (t) = V0 cos(2πft) I(t) I0 Ohm’s Law is valid at all times: V (t) t I(t) = R The current is then The current through a resistor is V0 in phase with the voltage. I(t) = cos(ωt) = I0 cos(ωt) R PHY1032S — AC Circuits Capacitor Circuit +Q V0 cos ωt C −Q Kirchhoff’s Voltage Law: Q(t) V (t) + VC (t) = V0 cos(ωt) − =0 C What is the current through the capacitor? ∆Q(t) ∆CV0 cos(ωt) ∆ cos(ωt) IC (t) = = = CV0 = −ωCV0 sin(ωt) ∆t ∆t ∆t PHY1032S — AC Circuits Capacitor Circuits +Q V (t) V0 cos ωt C V0 IC −Q t V (t) = V0 cos(ωt) Q(t) Q(t) = CV0 cos(ωt) CV0 Voltage and charge are in phase. t I(t) = ∆Q(t) = −ωCV0 sin(ωt) IC (t) ∆t ωCV0 Voltage and current of a capacitor are are out of phase. The current t precedes the voltage. PHY1032S — AC Circuits Capacitive Reactance We introduce the capacitive reactance V0 1 1 XC = = = IC0 ωC 2πfC We use the capacitive reactance to calculate amplitudes of V ,I in a capacitor. The unit is ohm (Ω). The capacitive reactance is similar to Ohm’s law, but not valid for instantaneous values V (t),I(t) frequency dependent: large f → small XC → large IC PHY1032S — AC Circuits Variable Current in Solenoid What happens when we switch on a current through a solenoid? change of current results in change of magnetic field / magnetic flux Lenz’ Law: induced voltage opposing change in current PHY1032S — AC Circuits Variable Current in Solenoid Magnitude of the induced voltage: ∆Φ ∆B ∆IL (t) VL (t) = N ∝ ∝ ∆t ∆t ∆t The induced voltage VL is proportional to ∆i/∆t, and therefore: ∆IL (t) VL (t) = L ∆t The proportionality factor L is called inductance and has the unit: 1 henry = 1H = 1Vs/A = 1Ωs PHY1032S — AC Circuits Inductor Circuit We have just seen ∆IL (t) V0 cos ωt L VL (t) = L IL ∆t We can guess - and check - the correct form for IL (t): IL (t) = I0 sin(ωt) ∆ sin(ωt) VL (t) = LI0 = ωLI0 cos(ωt) ∆t |{z} V0 This has the required form to match V (t) for V0 V0 = ωLI0 I0 = ωL PHY1032S — AC Circuits Inductor Circuit V0 cos ωt L V (t) IL V0 t VL (t) = V0 cos(ωt) V0 IL (t) IL (t) = sin(ωt) ωL V0 /ωL Voltage and current of a capacitor t are are out of phase. The current lags behind the voltage. PHY1032S — AC Circuits Inductive Reactance We can write the previous relation in a similar form as Ohm’s law: VL = 2πfL IL = IL XL VL IL = XL Where XL is the inductive reactance XL = 2πfL = 2πfL The inductive reactance is proportional to the frequency. The current in the circuit will thus depend on the frequency of the applied voltage. An inductor has high reactance (bad conductor) for high frequencies, and low reactance (good conductor) for low frequencies. PHY1032S — AC Circuits Resistor: Voltage and Current V (t) V0 cos ωt I R V0 t Input voltage: V (t) = V0 cos(2πft) I(t) I0 Ohm’s Law is valid at all times: V (t) t I(t) = R The current is then The current through a resistor is V0 in phase with the voltage. I(t) = cos(ωt) = I0 cos(ωt) R PHY1032S — AC Circuits Capacitor Circuits +Q V (t) V0 cos ωt C V0 IC −Q t V (t) = V0 cos(ωt) Q(t) Q(t) = CV0 cos(ωt) CV0 Voltage and charge are in phase. t I(t) = ∆Q(t) = −ωCV0 sin(ωt) IC (t) ∆t ωCV0 Voltage and current of a capacitor are are out of phase. The current t precedes the voltage. PHY1032S — AC Circuits Inductor Circuit V0 cos ωt L V (t) IL V0 t VL (t) = V0 cos(ωt) V0 IL (t) IL (t) = sin(ωt) ωL V0 /ωL Voltage and current of a capacitor t are are out of phase. The current lags behind the voltage. PHY1032S — AC Circuits AC Power in Resistors V (t) V0 t V (t) = V0 cos(ωt) V (t) V0 I(t) = = cos(ωt) = I0 cos(ωt) I(t) R R I0 P(t) = I(t) V (t) = I0 V0 cos2 (ωt) t The power is always positive → a resistor always dissipates power. P(t) Ppeak Ppeak = I0 V0 1 hPi = I0 V0 cos2 (2πft) = I0 V0 2 t PHY1032S — AC Circuits AC Power in Capacitors V (t) V0 V (t) = V0 cos(ωt) t V0 I(t) = − sin(ωt) = −I0 sin(ωt) ωC I(t) P(t) = I(t) V (t) = −I0 V0 cos(ωt) sin(ωt) I0 The power osscilates, and changes sign, with t twice the frequency of voltage and current. The capacitor accepts energy when the P(t) voltage is increasing, stores it and provides Ppeak the energy when the voltage is decreasing. On average, the capacitor neither dissipates t nor provides energy. PHY1032S — AC Circuits AC Power in Inductors V (t) V0 V (t) = V0 cos(ωt) t I(t) = ωLV0 sin(ωt) = I0 sin(ωt) I(t) P(t) = I(t) V (t) = I0 V0 cos(ωt) sin(ωt) I0 The power osscilates, and changes sign, with twice the frequency of voltage and current. t The inductor accepts energy when the P(t) current is increasing, stores it and provides Ppeak the energy when the current is decreasing. On average, the inductor neither dissipates t nor provides energy. PHY1032S — AC Circuits LC Circuit Kirchhoff’s Junction Law: IL (t) = IC (t) = I(t) Kirchhoff’s Loop Law: ∆I Q VL (t) + VC (t) = L + =0 ∆t C PHY1032S — AC Circuits Comparison: Harmonic Oscillator and LC Circuit The equations for Q and I in the LC circuit look very similar to the equations of the simple harmonic oscillator: Q x= ? VC = = ? C ∆x ∆Q v= I= ∆t ∆t ∆v ∆I a= VL = L ∆t ∆t The key for harmonic motion is that the first and the third equation are linked: F kx a= =− VL = −VC m m PHY1032S — AC Circuits Reminder: Solution for the Simple Harmonic Oscillator The simple harmonic oscillator is described by the equations: ∆x ∆v x= ? v= a= ∆t ∆t F kx a= =− m m We know the solutions of these equations from the first semester: x(t) = x0 cos(2πft) or x(t) = x0 sin(2πft) or a superposition of the two. We can also calculate the frequency: We know from the harmonic oscillator: m m d 2x 1 m r x =− a=− f = k k dt 2 |{z} 2π k (2πf )2 PHY1032S — AC Circuits Oscillations of LC Circuits PHY1032S — AC Circuits Energy in LC Circuits We know from the harmonic oscillator: m m d 2x x =− a=− k k dt 2 |{z} (2πf )2 The LC circuit is the electric equivalent of an oscillator: d 2 VC VC = − |{z} LC 2 dt 2 ω By comparison, we get for the LC circuit 1 1 f = √ ω=√ 2π LC LC PHY1032S — AC Circuits Driven RLC Circuit What happens if we connect a resistor, a capacitor and an inductor in series to an AC voltage source? PHY1032S — AC Circuits Kirchhoff’s Laws Kirchhoff’s Junction Law: