Induction and Inductance

Induction and Inductance

Faraday’s Law and Lenz’s Law
First Experiment. Figure shows a conducting loop connected to a sensitive ammeter. Because there is no battery or other source of emf included, there is no current in the circuit. However, if we move a bar magnet toward the loop, a current suddenly appears in the circuit. The current disappears when the magnet stops moving. If we then move the magnet away, a current again suddenly appears, but now in the opposite direction. If we experimented for a while, we would discover the following: A current appears only if there is relative motion between the loop and the magnet (one must move relative to the other); the current disappears when the relative motion between them ceases. Faster motion of the magnet produces a greater current. If moving the magnet’s north pole toward the loop causes, say, clockwise current, then moving the north pole away causes counterclockwise current. Moving the south pole toward or away from the loop also causes currents, but in the reversed directions from the north pole effects.

Second Experiment. For this experiment we use the apparatus shown in the figure, with the two conducting loops close to each other but not touching. If we close switch S to turn on a current in the right-hand loop, the meter suddenly and briefly registers a current—an induced current—in the left-hand loop. If the switch remains closed, no further current is observed. If we then open the switch, another sudden and brief induced current appears in the left-hand loop, but in the opposite direction. We get an induced current (from an induced emf) only when the current in the right-hand loop is changing (either turning on or turning off) and not when it is constant (even if it is large). The induced emf and induced current in these experiments are apparently caused when something changes — but what is that “something”? Faraday knew.

Faraday realized that an emf E and a current can be induced in a loop, as in our two experiments, by changing the amount of magnetic field passing through the loop. He further realized that the “amount of magnetic field” can be visualized in terms of the magnetic field lines passing through the loop. The magnetic flux ΦB through an area A in a magnetic field B is defined as where the integral is taken over the area. The SI unit of magnetic flux is the weber, where 1 Wb = 1 T  m2. If B is perpendicular to the area and uniform over it, the flux is

Faraday’s Law of Induction Faraday’s Law. With the notion of magnetic flux, we can state Faraday’s law in a more quantitative and useful way: the induced emf tends to oppose the flux change and the minus sign indicates this opposition. This minus sign is referred to as Lenz’s Law.

Concept Check – Magnetic Flux
In order to change the magnetic flux through the loop, what would you have to do? 1. drop the magnet 2. move the magnet upward 3. move the magnet sideways 4. only (1) and (2) 5. all of the above

In order to change the magnetic flux through the loop, what would you have to do? 1. drop the magnet 2. move the magnet upward 3. move the magnet sideways 4. only (1) and (2) 5. all of the above Moving the magnet in any direction would change the magnetic field through the loop and thus the magnetic flux.

In order to change the magnetic flux through the loop, what would you have to do? 1. tilt the loop 2. change the loop area 3. use thicker wires 4. only (1) and (2) 5. all of the above

In order to change the magnetic flux through the loop, what would you have to do? 1. tilt the loop 2. change the loop area 3. use thicker wires 4. only (1) and (2) 5. all of the above Since F = B A cosq , changing the area or tilting the loop (which varies the projected area) would change the magnetic flux through the loop. The units of magnetic flux are tesla-meter2, which is called a weber (Wb).

Concept Check – Induced Voltage
Wire 1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire 2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced voltages in these two cases. 1. V1 > V2 2. V1 < V2 3. V1 = V2  0 4. V1 = V2 = 0 N S 1 2

Concept Check – Induced Voltage
Wire 1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire 2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced voltages in these two cases. 1. V1 > V2 2. V1 < V2 3. V1 = V2  0 4. V1 = V2 = 0 Faraday’s Law: depends on N (number of loops) so the induced emf is twice as large in the wire with 2 loops. N S 1 2 A single loop of wire will experience an induced voltage that equals the time rate of change of magnetic flux through it an any given instant. If there are N turns of wire in a coil, each has an induced voltage that is in series with the others so that the net induced emf is given by the equation above. The negative sign relates the polarity of the induced emf to the flux change. The induced emf will produce a current that always acts to oppose the change that originally caused it. This is Lenz’s law. An approaching N pole induces a current that causes the coil to manifest an S pole at its near end, which opposes the advance of the magnet and thereby the change of the field.

Concept Check – Induced Current
Wire 1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire 2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases. 1. I1 > I2 2. I1 < I2 3. I1 = I2  0 4. I1 = I2 = 0 N S 1 2

Concept Check – Induced Current
Wire 1 (length L) forms a one-turn loop, and a bar magnet is dropped through. Wire 2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases. 1. I1 > I2 2. I1 < I2 3. I1 = I2  0 4. I1 = I2 = 0 Faraday’s law: says that the induced emf is twice as large in the wire with 2 loops. The current is given by Ohm’s law: I = V/R. Since Wire 2 is twice as long as Wire 1, it has twice the resistance, so the current in both wires is the same. N S 1 2

Sample 30.01

An induced current has a direction such that the magnetic field due to this induced current opposes the change in the magnetic flux that induces the current. The induced emf has the same direction as the induced current. Lenz’s law at work. As the magnet is moved toward the loop, a current is induced in the loop. The current produces its own magnetic field, with magnetic dipole moment μ oriented so as to oppose the motion of the magnet. Thus, the induced current must be counterclockwise as shown.

The direction of the current i induced in a loop is such that the current’s magnetic field Bind opposes the change in the magnetic field B inducing i. The field Bind is always directed opposite an increasing field B (a, c) and in the same direction as a decreasing field B (b, d ). The curled – straight right-hand rule gives the direction of the induced current based on the direction of the induced field. The direction of the current i induced in a loop is such that the current’s magnetic field Bind opposes the change in the magnetic field B inducing i. The field Bind is always directed opposite an increasing field B (a, c) and in the same direction as a decreasing field B (b, d ). The curled – straight right-hand rule gives the direction of the induced current based on the direction of the induced field.

Concept Check – Electromagnetic Induction (2)
If a North pole moves toward the loop in the plane of the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current

Concept Check – Electromagnetic Induction (2)
If a North pole moves toward the loop in the plane of the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current Since the magnet is moving parallel to the loop, there is no magnetic flux through the loop. Thus the induced current is zero.

Concept Check – Shrinking Wire Loop
If a coil is shrinking in a magnetic field pointing into the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current

Concept Check – Shrinking Wire Loop
If a coil is shrinking in a magnetic field pointing into the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current The magnetic flux through the loop is decreasing, so the induced B field must try to reinforce it and therefore points in the same direction—into the page. According to the right-hand rule, an induced clockwise current will generate a magnetic field into the page.

Concept Check – Rotating Wire Loop
If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current

Concept Check – Rotating Wire Loop
If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current As the coil is rotated into the B field, the magnetic flux through it increases. According to Lenz’s Law, the induced B field has to oppose this increase, thus the new B field points to the right. An induced counterclockwise current produces just such a B field.

Concept Check – Electromagnetic Induction
If a North pole moves toward the loop from above the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current

Concept Check – Electromagnetic Induction
If a North pole moves toward the loop from above the page, in what direction is the induced current? 1. clockwise 2. counterclockwise 3. no induced current The magnetic field of the moving bar magnet is pointing into the page and getting larger as the magnet moves closer to the loop. Thus the induced magnetic field has to point out of the page. A counterclockwise induced current will give just such an induced magnetic field. Follow-up: What happens if the magnet is stationary but the loop moves?

Concept Check – Moving Wire Loop
A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1. clockwise 2. counterclockwise 3. no induced current x x x x x x x x x x x x

Concept Check – Moving Wire Loop
A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1. clockwise 2. counterclockwise 3. no induced current Since the magnetic field is uniform, the magnetic flux through the loop is not changing. Thus NO current is induced. x x x x x x x x x x x x Follow-up: What happens if the loop moves out of the page?

Concept Check – Moving Wire Loop (2)
A wire loop is being pulled through a uniform magnetic field that suddenly ends. What is the direction of the induced current? 1. clockwise 2. counterclockwise 3. no induced current x x x x x

A wire loop is being pulled through a uniform magnetic field that suddenly ends. What is the direction of the induced current? 1. clockwise 2. counterclockwise 3. no induced current The B field into the page is disappearing in the loop, so it must be compensated by an induced flux also into the page. This can be accomplished by an induced current in the clockwise direction in the wire loop. x x x x x

What is the direction of the induced current if the B field suddenly increases while the loop is in the region? 1. clockwise 2. counterclockwise 3. no induced current x x x x x x x x x x x x x x x x x

What is the direction of the induced current if the B field suddenly increases while the loop is in the region? 1. clockwise 2. counterclockwise 3. no induced current The increasing B field into the page must be countered by an induced flux out of the page. This can be accomplished by induced current in the counterclockwise direction in the wire loop. x x x x x x x x x x x x

Concept Check – Loop and Wire
A wire loop is being pulled away from a current-carrying wire. What is the direction of the induced current in the loop? 1. clockwise 2. counterclockwise 3. no induced current I

Concept Check – Loop and Wire
A wire loop is being pulled away from a current-carrying wire. What is the direction of the induced current in the loop? 1. clockwise 2. counterclockwise 3. no induced current The magnetic flux is into the page on the right side of the wire and decreasing due to the fact that the loop is being pulled away. By Lenz’s Law, the induced B field will oppose this decrease. Thus, the new B field points into the page, which requires an induced clockwise current to produce such a B field. I

Concept Check – Loop and Wire (2)
What is the induced current if the wire loop moves in the direction of the arrow ? 1. clockwise 2. counterclockwise 3. no induced current I

Concept Check – Loop and Wire (2)
What is the induced current if the wire loop moves in the direction of the arrow ? 1. clockwise 2. counterclockwise 3. no induced current The magnetic flux through the loop is not changing as it moves parallel to the wire. Therefore, there is no induced current. I

Sample 30.02

Sample 30.03

Induction and Energy Transfer
By Lenz’s law, whether you move the magnet toward or away from the loop, a magnetic force resists the motion, requiring your applied force to do work. At the same time, thermal energy is produced in the material. The energy you transfer to the closed loop & magnet system ends up as thermal energy. The faster you move the magnet, the greater rate at which you do work (power).

In this figure, a rectangular loop of wire of width L has one end in a uniform external magnetic field that is directed perpendicularly into the plane of the loop. This field may be produced, for example, by a large electromagnet. The dashed lines in the figure show the assumed limits of the magnetic field; the fringing of the field at its edges is neglected. You are to pull this loop to the right at a constant velocity v.

Flux change: Therefore, in the figure a magnetic field and a conducting loop are in relative motion at speed v and the flux of the field through the loop is changing with time (here the flux is changing as the area of the loop still in the magnetic field is changing). Rate of Work: To pull the loop at a constant velocity v, you must apply a constant force F to the loop because a magnetic force of equal magnitude but opposite direction acts on the loop to oppose you. The rate at which you do work — that is, the power — is then where F is the magnitude of the force you apply to the loop.

Induced emf: To find the current, we first apply Faraday’s law. When x is the length of the loop still in the magnetic field, the area of the loop still in the field is Lx. Then, the magnitude of the flux through the loop is As x decreases, the flux decreases. Faraday’s law tells us that with this flux i decrease, an emf is induced in the loop. We can write the magnitude of this emf as in which we have replaced dx/dt with v, the speed at which the loop moves. A circuit diagram for the loop of above figure while the loop is moving.

Induced Current: Figure (bottom) shows the loop as a circuit: induced emf is represented on the left, and the collective resistance R of the loop is represented on the right. To find the magnitude of the induced current, we can apply the equation which gives In the Fig. (top), the deflecting forces acting on the three segments of the loop are marked F1, F2, and F3. Note, however, that from the symmetry, forces F2 and F3 are equal in magnitude and cancel. This leaves only force F1, which is directed opposite your force F on the loop and thus is the force opposing you. A circuit diagram for the loop of above figure while the loop is moving.

So, F = -F1. the magnitude of F1 thus (from ) where the angle between B and the length vector L for the left segment is 90°. This gives us Because B, L, and R are constants, the speed v at which you move the loop is constant if the magnitude F of the force you apply to the loop is also constant. Rate of Work: We find the rate at which you do work on the loop as you pull it from the magnetic field: A circuit diagram for the loop of above figure while the loop is moving. NOTE: The work that you do in pulling the loop through the magnetic field appears as thermal energy in the loop.

Concept Check – Falling Magnet
A bar magnet is held above the floor and dropped. In 1, there is nothing between the magnet and the floor. In 2, the magnet falls through a copper loop. How will the magnet in case 2 fall in comparison to case 1? 1. it will fall slower 2. it will fall faster 3. it will fall the same copper loop N S 2 1

Concept Check – Falling Magnet
A bar magnet is held above the floor and dropped. In 1, there is nothing between the magnet and the floor. In 2, the magnet falls through a copper loop. How will the magnet in case 2 fall in comparison to case 1? 1. it will fall slower 2. it will fall faster 3. it will fall the same When the magnet is falling from above the loop in 2, the induced current will produce a North pole on top of the loop, which repels the magnet. When the magnet is below the loop, the induced current will produce a North pole on the bottom of the loop, which attracts the South pole of the magnet. copper loop N S 2 1 The behavior of the magnet can also be understood in terms of energy conservation. The downward motion of the magnet produces a counterforce that it must overcome to continue to fall. The work done in the process (by gravity or some other force) is converted into electrical energy of the charge in the wire causing it to move in the form of electrical current. Current cannot be created at no cost…the energy has to come from somewhere. No matter how the flux changes, work must be done on the system and the induced current, which is a manifestation of the work done, must oppose the change. The work done by the magnet on the current must match the work done by the current on the magnet, and that work is negative.

Concept Check – Falling Magnet (2)
If there is induced current, doesn’t that cost energy? Where would that energy come from in case 2? 1. induced current doesn’t need any energy 2. energy conservation is violated in this case 3. there is less KE in case 2 4. there is more gravitational PE in case 2 copper loop N S 2 1

Concept Check – Falling Magnet (2)
If there is induced current, doesn’t that cost energy? Where would that energy come from in case 2? 1. induced current doesn’t need any energy 2. energy conservation is violated in this case 3. there is less KE in case 2 4. there is more gravitational PE in case 2 In both cases, the magnet starts with the same initial gravitational PE. In case 1, all the gravitational PE has been converted into kinetic energy. In case 2, we know the magnet falls slower, thus there is less KE. The difference in energy goes into making the induced current. copper loop N S 2 1

Induced Electric Field
(a) If the magnetic field increases at a steady rate, a constant induced current appears, as shown, in the copper ring of radius r. (b) An induced electric field exists even when the ring is removed; the electric field is shown at four points. (c) The complete picture of the induced electric field, displayed as field lines. (d) Four similar closed paths that enclose identical areas. Equal emfs are induced around paths 1 and 2, which lie entirely within the region of changing magnetic field. A smaller emf is induced around path 3, which only partially lies in that region. No net emf is induced around path 4, which lies entirely outside the magnetic field.

Therefore, an emf is induced by a changing magnetic flux even if the loop through which the flux is changing is not a physical conductor but an imaginary line. The changing magnetic field induces an electric field E at every point of such a loop.

Using the induced electric field, we can write Faraday’s law in its most general form as Electric Potential: Induced electric fields are produced not by static charges but by a changing magnetic flux. Therefore,

Sample 30.04

Inductors and Inductance
An inductor is a device that can be used to produce a known magnetic field in a specified region. If a current i is established through each of the N windings of an inductor, a magnetic flux ΦB links those windings. The inductance L of the inductor is The SI unit of inductance is the henry (H), where 1 henry = 1H = 1Tm2/A. The windings of the inductor are said to be linked by the shared flux and the product N is said to be the magnetic flux linkage. Inductance is therefore a measure of the flux linkage produced by the inductor per unit of current. The crude inductors with which Michael Faraday discovered the law of induction. In those days amenities such as insulated wire were not commercially available. It is said that Faraday insulated his wires by wrapping them with strips cut from one of his wife’s petticoats.

Inductors and Inductance
Consider a long solenoid of cross-sectional area A. The flux linkage of a length l near the middle is From Ch. 29, the magnitude B is given by The inductance per unit length near the middle of a long solenoid of cross-sectional area A and n turns per unit length is The crude inductors with which Michael Faraday discovered the law of induction. In those days amenities such as insulated wire were not commercially available. It is said that Faraday insulated his wires by wrapping them with strips cut from one of his wife’s petticoats.

Self-Induction If two coils — which we can now call inductors — are near each other, a current i in one coil produces a magnetic flux ΦB through the second coil. We have seen that if we change this flux by changing the current, an induced emf appears in the second coil according to Faraday’s law. An induced emf appears in the first coil as well. This process (see Figure) is called self-induction, and the emf that appears is called a self-induced emf. It obeys Faraday’s law of induction just as other induced emfs do. For any inductor, Faraday’s law tells us that By combining these equations, we can write Note: Thus, in any inductor (such as a coil, a solenoid, or a toroid) a self-induced emf appears whenever the current changes with time. The magnitude of the current has NO influence on the magnitude of the induced emf; only the rate of change of the current counts.

Self-Induction

RL Circuits 𝑖= 𝑑𝑞 𝑑𝑡 =− 𝑞 0 𝑅𝐶 𝑒 − 𝑡 𝑅𝐶 (discharging a capacitor)
For an RC circuit: 𝑖= 𝑑𝑞 𝑑𝑡 =− 𝑞 0 𝑅𝐶 𝑒 − 𝑡 𝑅𝐶 (discharging a capacitor) In Module 27-4 we saw that if we suddenly introduce an emf into a single-loop circuit containing a resistor R and a capacitor C, the charge on the capacitor does not build up immediately to its final equilibrium value but approaches it in an exponential fashion:

RL Circuits An analagous slowing of the rise (or fall) or a current occurs if we introduce an emf into (or remove it from) a single loop circuit containing a resistor and an inductor. If a constant emf is introduced into a single-loop circuit containing a resistance R and an inductance L, the current rises to an equilibrium value of /R according to Here tL, the inductive time constant, is given by

RL Circuits The time constant 𝜏 is the time it takes the current in the circuit to reach about 63% of its final equilibrium value 𝜀 𝑅 .

RL Circuits Plot (a) and (b) shows how the potential differences VR (= iR) across the resistor and VL (= L di/dt) across the inductor vary with time for particular values of , L, and R. When the source of constant emf is removed and replaced by a conductor, the current decays from a value i0 according to The variation with time of (a) VR, the potential difference across the resistor in the circuit (top), and (b) VL, the potential difference across the inductor in that circuit.

Sample 30.05

Sample 30.06

Energy Stored in a Magnetic Field
Applying the loop rule to the LR circuit, An RL circuit.

Sample 30.07

Energy Density of a Magnetic Field
Consider a length l near the middle of a long solenoid of cross-sectional area A carrying current i; the volume associated with this length is Al. The energy UB stored by the length l of the solenoid must lie entirely within this volume because the magnetic field outside such a solenoid is approximately zero. Moreover, the stored energy must be uniformly distributed within the solenoid because the magnetic field is (approximately) uniform everywhere inside. Thus, the energy stored per unit volume of the field is We have, here L is the inductance of length l of the solenoid Substituting for L/l we get And we can write the energy density as

Mutual Induction Mutual induction. (a) The magnetic field B1 produced by current i1 in coil 1 extends through coil 2. If i1 is varied (by varying resistance R), an emf is induced in coil 2 and current registers on the meter connected to coil 2. (b) The roles of the coils interchanged. If coils 1 and 2 are near each other, a changing current in either coil can induce an emf in the other. This mutual induction is described by and

Summary Magnetic Flux Lenz’s Law Emf and the Induced Magnetic Field
The magnetic flux through an area A in a magnetic field B is defined as If B: is perpendicular to the area and uniform over it, Eq becomes Lenz’s Law An induced current has a direction such that the magnetic field due to this induced current opposes the change in the magnetic flux that induces the current. Eq. 30-1 Emf and the Induced Magnetic Field The induced emf is related to E by Faraday’s law in its most general form, Eq. 30-2 Faraday’s Law of Induction The induced emf is, If the loop is replaced by a closely packed coil of N turns, the induced emf is 3 Eq Eq. 30-4 Eq Eq. 30-5

Summary Inductor Series RL Circuit Magnetic Energy Self-Induction
The inductance L of the inductor is The inductance per unit length near the middle of a long solenoid of cross-sectional area A and n turns per unit length is Series RL Circuit Rise of current, Decay of current Eq Eq Eq Magnetic Energy the inductor’s magnetic field stores an energy given by The density of stored magnetic energy, Eq Self-Induction This self-induced emf is, 3 Eq Eq Mutual Induction The mutual induction is described by, Eq Eq Eq

Gauss’ Law for Magnetic Fields
Gauss’ law for magnetic fields is a formal way of saying that magnetic monopoles do not exist. The law asserts that the net magnetic flux ΦB through any closed Gaussian surface is zero: Contrast this with Gauss’ law for electric fields, The field lines for the magnetic field B of a short bar magnet. The red curves represent cross sections of closed, three-dimensional Gaussian surfaces. If you break a magnet, each fragment becomes a separate magnet, with its own north and south poles. Gauss’ law for magnetic fields says that there can be no net magnetic flux through the surface because there can be no net “magnetic charge” (individual magnetic poles) enclosed by the surface.

Induced Magnetic Fields
A changing electric flux induces a magnetic field B. Maxwell’s Law, Relates the magnetic field induced along a closed loop to the changing electric flux ϕE through the loop. Charging a Capacitor. As an example of this sort of induction, we consider the charging of a parallel-plate capacitor with circular plates. The charge on our capacitor is being increased at a steady rate by a constant current i in the connecting wires. Then the electric field magnitude between the plates must also be increasing at a steady rate.

A changing electric flux induces a magnetic field B. Maxwell’s Law, Relates the magnetic field induced along a closed loop to the changing electric flux ϕE through the loop. Charging a Capacitor (continued) Figure (b) is a view of the right-hand plate of Fig. (a) from between the plates. The electric field is directed into the page. Let us consider a circular loop through point 1 in Figs.(a) and (b), a loop that is concentric with the capacitor plates and has a radius smaller than that of the plates. Because the electric field through the loop is changing, the electric flux through the loop must also be changing. According to the above equation, this changing electric flux induces a magnetic field around the loop.

Ampere-Maxwell Law Ampere’s law, gives the magnetic field generated by a current ienc encircled by a closed loop. Thus, the two equations (the other being Maxwell’s Law) that specify the magnetic field B produced by means other than a magnetic material (that is, by a current and by a changing electric field) give the field in exactly the same form. We can combine the two equations into the single equation: When there is a current but no change in electric flux (such as with a wire carrying a constant current), the first term on the right side of Eq. is zero, and so the Eq. reduces to Ampere’s law. When there is a change in electric flux but no current (such as inside or outside the gap of a charging capacitor), the second term on the right side of Eq. is zero, and so Eq. reduces to Maxwell’s law of induction.

Displacement Current If you compare the two terms on the right side of Eq. (Ampere-Maxwell Law), you will see that the product ε0(dϕE/dt) must have the dimension of a current. In fact, that product has been treated as being a fictitious current called the displacement current id: Ampere-Maxwell Law then becomes, where id,enc is the displacement current encircled by the integration loop. Ampere-Maxwell law

Displacement Current Finding the Induced Magnetic Field: In Chapter 29 we found the direction of the magnetic field produced by a real current i by using the right-hand rule. We can apply the same rule to find the direction of an induced magnetic field produced by a fictitious displacement current id, as is shown in the center of Fig. (c) for a capacitor. Then, as done previously, the magnitude of the magnetic field at a point inside the capacitor at radius r from the center is the magnitude of the magnetic field at a point outside the capacitor at radius r is (a) Before and (d) after the plates are charged, there is no magnetic field. (b) During the charging, magnetic field is created by both the real current and the (fictional) (c) displacement current. (c) The same right-hand rule works for both currents to give the direction of the magnetic field.

Displacement Current The four fundamental equations of electromagnetism, called Maxwell’s equations and are displayed in Table 32-1. These four equations explain a diverse range of phenomena, from why a compass needle points north to why a car starts when you turn the ignition key. They are the basis for the functioning of such electromagnetic devices as electric motors, television transmitters and receivers, telephones, scanners, radar, and microwave ovens.

Induction and Inductance

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