Electromagnetic Induction

Electromagnetic induction

So far we have found that v kQ Electric charges create E-fields E = rˆ r 2 v v µ qv × rˆ Moving electric charges create B-fields B = 0 4π r 2 But this is not the whole story!

Electromagnetic induction

Electric Fields can also be created by a changing Magnetic Field.

This is described by “Faraday’s Law”.

Magnetic Fields can also be created by a changing Electric Field.

This will help complete the set of equations called Maxwell’s equations of electricity and magnetism.

Electricity and Magnetism can be unified in a single theory! The electricity and magnetic forces are manifestations of a single “electromagnetic interaction”

2 Electromagnetic induction

Magnetic Flux Magnetic Flux

The flux through an element of area is:

To calculate the flux through the surface, we need to integrate:

B For the simple case of a flat surface, we obtain: φ

Magnetic flux Clicker Question At what angle of orientation of the surface is the flux through the surface half of the maximum possible flux?

A) φ = 0 degrees B B) φ = 30 degrees φ C) φ = 45 degrees D) φ = 60 degrees E) None of the above

Faraday’s Law of Induction

A changing (time-dependent) magnetic field “induces” (generates) an EMF ε. The magnitude of the EMF equals minus the rate of change of the flux. dΦ ε = − B dt Two new quantities introduced. 1. Magnetic Flux = ΦB 2. Electro Motive Force (EMF) = ε

8 Important observations:

• Recall: EMF is not a force, but a source of voltage capable of generating power. • The EMF is called “motional” EMF, to distinguish from “chemical” EMF (batteries) • Recall: The magnetic flux through a closed surface is zero! (Gauss’ Law for B-fields) • The surface for Faraday’s Law is an open surface!

Clicker Question We have a square imaginary loop with side of length a. There is a Magnetic Field that is uniform throughout the region as shown. How does the Magnetic Flux magnitude

|ΦB| change if we halve the Magnetic Field strength and double the sides of the square loop? A) Flux goes to ½ original value B) Flux goes to ¼ original value X B C) Flux goes to 2 times original value D) Flux does not change E) None of the above

10 Changing the magnetic flux

1. Change the strength of the Magnetic Field

2. Change the area of the loop

3. Change the orientation of the loop (A vector) and the B-field

Clicker Question

A loop of wire is moving rapidly through a uniform magnetic field as shown. Is a non-zero EMF induced in the loop?

A) No, there in no EMF B) Yes, there is an EMF

12 Clicker Question A loop of wire is spinning rapidly about a stationary axis in uniform magnetic field as shown. True or False: There is a non-zero EMF induced in the loop.

A) False, zero EMF B) True, there is an induced EMF

F = BAcosφ and the angle φ is changing.

Direction of the EMF around a loop Lenz’s Law The induced EMF tends to induce a current in the direction that opposes the changes in magnetic flux

d Φ ε = − B dt

The (-) negative sign in Faraday’s Law is more of a reminder. Lenz’s Law is just another way to express Faraday’s Law. Moreover, it can be deduced from it.

Lenz’s Law If we move the magnet closer to the loop, what is the direction of the induced EMF and thus the direction of the current in the loop? v v 1. The Magnetic Φ B = ∫ B ⋅dA Flux is increasing. surf dΦ 2. Therefore B > 0 dt 3. Induced current must create B-field that fights the change!

16 Lenz’s Law

dΦ B > 0 i dt B Induced current must create B-field that fights the change!

An induced B-field down through the loop

will create a slight decrease in ΦB.

Thus, we have determined the direction of the induced current and EMF.

Lenz’s Law

“The Magnetic Flux ΦB Up Through the Loop is Increasing.

To Fight this Change, an Induced B-field is made that Decreases the Flux Up Through the Loop.

This is accomplished by the induced current.”

18 Lenz’s law (summary)

Clicker Question

A bar magnet is positioned below a loop of wire. The magnet is pulled down, away from the loop. As viewed from above, is the induced current in the loop clockwise, counterclockwise, or zero? B eyeball i A: clockwise B: counter-clockwise

C: zero

Answer: Counterclockwise. As the magnet is pulled away, the flux is decreasing. To fight the decrease, the induced B-field should add to the original B-field.

20 Clicker Question

A loop of wire is sitting in a uniform, constant magnet field as shown. Suddenly, the loop is bent into a smaller area loop. During the bending of the loop, the induced current in the loop is ... A: zero B: clockwise C: counterclockwise

B(in) B(in)

Answer: Clockwise. The flux into the page is decreasing as the loop area decreases. To fight the decrease, we want the induced B to add to the original B. By the right hand rule (version II) , a clockwise induced current will make an induced B into the page, adding to the original B. 21

Clicker Question

Is there an induced current in this circuit? If so, what is its direction?

A. Yes, clockwise B. Yes, counterclockwise C. No

* There is zero Magnetic Flux at all times through the loop.

22 Clicker Question A current-carrying wire is pulled away from a conducting loop in the direction shown. As the wire is moving, is there a clockwise current around the loop, a counterclockwise current or no current?

A. There is a clockwise current around the loop. B. There is a counterclockwise current around the loop. C. There is no current around the loop.

2 2 Example with numbers: Area = (0.1m) =0.01 m dB/dt = +0.1 Tesla/second

B (increasing) dΦ d dB | ε |= B = (BA) = A dt dt dt cm 0 1 | ε |= (0.01)(0.1) = 0.001Volts Suppose there are 1000 loops to the wire coil. V 10 cm dΦ (N) dΦ (1) | ε |= B = N B =1Volt dt dt

24 Generators

Generators use Faraday’s Law to convert mechanical energy into electrical energy.

This is the opposite of motors which convert electrical into mechanical energy.

The alternator

• Wire loop in a constant external B-field. • Turning a crank makes the loop rotate. • This then induces a current in the wire loop.

B B X X i (induced)

26 The alternator

• Wire loop in a constant external B-field. • Turning a crank makes the loop rotate. • This then induces a current in the wire loop.

Clicker Question Instead of turning a crank and inducing a current, what if you hook up the wire loop to a battery and push current through it.

What happens to the loop? B A) Nothing X B i B) It feels a net force X C) It feels a net torque D) The current stops flowing

Note that the wire loop is a rectangle that is slightly rotated from the plane of the page.

28 Generator Motor Converts mechanical energy Converts electrical energy into into electrical energy. mechanical energy.

Crank or otherwise Battery creates a current in a mechanically turns a wire loop loop. This interacts with an in an external B-field. This external B-field to create a induces an electric current in mechanical torque. the loop.

The “sliding bar generator”

Sliding metal bar on metal rails Bar is pulled by an external agent to the B v right at constant L X X X X X X velocity.

There is a uniform B- field surrounding the x=vt rails and loop system.

30 The “sliding bar generator”

This creates a wire loop B v whose area is growing with L X X X X X X time. A = Lx = Lvt x=vt Thus, the Magnetic Flux is increasing with time! d d d Φ = (BA) = B (Lvt) = BLv dt B dt dt

Clicker Question

What is the direction of the induced EMF and B v thus current in the wire L X X X X X X loop?

A) Clockwise B) Counter Clockwise x=vt Motional EMF induced ε = BLv

32 The “sliding bar generator”

There is another way to see why the current in the bar is up.

i Charges in the bar are moving to the right with the bar itself.

There is a B-field in this region. v Thus, there is an upward force on the charges. v v v X F = qv × B = qvB (up) B Thus, there is an upward current in the bar!

The “sliding bar generator”

One more consequence…. If there is an upward current in the wire and there i FM FAgent is an external B-field, the we get? v v v

FMag = IL× B = ILB (left) v But, there is also the external agent pulling the rod to the right. X If the velocity is constant, what is true about the B forces? They must be equal and opposite!

34 Clicker Question A conducting loop is halfway into a magnetic field. Suppose the magnetic field begins to increase rapidly in strength. What happens to the loop?

A. The loop is pushed upward, toward the top of the page. B. The loop is pushed downward, toward the bottom of the page. C.The loop is pulled to the left, into the magnetic field. D.The loop is pushed to the right, out of the magnetic field. E. The tension is the wires increases but the loop does not move.

Eddy currents

If a metal object (something in which electric charges can move) and an external source of Magnetic Field are in relative motion so that there is changing Magnetic Flux through the metal, then Faraday says there is an EMF and thus an Eddy current.

If the metal is moving, then the direction of the current is always such as to cause a force which opposes the motion (slows the metal). Eddy currents: Applications

Eddy Current Brakes on Trains

Since there is no direct contact, there is minimal friction. Thus the brakes do not wear down!

Eddy currents: Applications

Metal detector Portable metal detector Eddy currents: Applications

Induction Cooktop

Works via changing B-field, inducing a current in the pot or wok (which must be metal). This current then leads to power loss as heating of the pot. The cooktop surface which is not metal and does not conduct, does not get hot.

Induced electric field

In the case where the loop or metal is stationary and the Magnetic field is changing, then there is no Magnetic force on the charges in the wire (since v=0) v v v FM = qv × B Thus, the only way to explain the induced current in the wire is to admit that there is the creation of a new Electric Field!

40 Motional EMF revisited Technically, the EMF around a closedv Loopv is defined as: ε = ∫ E ⋅dl Loop But, recall that Voltage is defined as: v v ∆V = −∫ E ⋅dl Was not this integral around a closed loop always zero? The motional EMF is not always zero around a closed loop !

Faraday’s Law reformulated

d Φ ε = − B dt

v v d v v ∫ E ⋅ d l = − ∫ B ⋅ d A Loop dt surface

How do the loop and the surface relate?

42 Motional EMF revisited

v v d v v ∫ E ⋅ d l = − ∫ B ⋅ d A Loop dt surf

Bexternal

v d v R 2 E ( 2 π R ) = − []B ext (π R ) dt

Motional EMF revisited

Wire Loop If B = constant, Then Magnetic Flux = constant

dΦB/dt = 0 EMF = 0 and Voltmeter reads 0. V B If B =increasing, Then Magnetic Flux = changing

|dΦB/dt| > 0 EMF > 0 and Voltmeter reads > 0.

What if the wire loop was not there?

44 Induced electric field

A changing B-field creates a new kind of E-field.

If B is decreasing E-field is CCW B E

B If B is increasing E-field is Clockwise E

Induced electric field

The (non-Coulomb) E-field is very different from the Coulomb E- field created by single electric charges.

E B q E

v v v v ∆V = −∫ E ⋅dl = 0 ε = ∫ E ⋅ dl ≠ 0 Loop

46 Time dependent B-fields So far we have seen that a time varying flux can be generated by moving a magnet, or by mechanically changing the surface area…

But what about changing the magnitude of the field?

From Ampere’s Law:

Replacing in Faraday’s Law we obtain:

Videos

http://www.youtube.com/watch?v=jBrneA-wQGA Tesla Ampere’s Law revisited

Maxwell realized that Ampere’s Law must be incomplete because he discovered a situation where it gave the wrong v v answer!

∫ B⋅dl = µ0ienc + ?

Ampere’s Law revisited

Consider a charging capacitor. The charge builds up in the plates, and there is a time-dependent current flowing through the wires I(t)

E B + - B + - + - + - + - Current I(t) Current I(t)

1. As the currents are flowing, one has Magnetic Fields around the wires. 2. Charge is also building up on the plates, creating an increasing Electric Field between them.

50 Clicker Question + - + - + - + - Current I Current I + -

B=? v v v µ I dL × rˆ B = ∫ dB = 0 ∫ tot 4π r2 Consider the Biot-Savart Law for Magnetic Fields. Is there a Magnetic Field at the point labeled between the plates? A)Yes, there is a B-Field B)No, there is zero B-field

Clicker Question + - + - + - + - Current I Current I + - B=? Now consider an Amperian Loop as drawn. According to Ampere’s Law is there a Magnetic Field at the point labeled between the plates? A)Yes, there is a B-Field Something must be wrong with B)No, there is zero B-field Ampere’s Law!

52 Ampere’s Law revisited

+ - + - + - + - Current I Current I + -

Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose.

Ampere’s Law revisited

Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose. Ampere’s law: solving the paradox

The changing Electric Field inside the capacitor must act like a current enclosed for Ampere’s Law. v v ⎛ dΦ ⎞ ∫ B ⋅dl = µ ⎜ I + ε E ⎟ 0 ⎝ enc 0 dt ⎠ v v Where Φ E = ∫ E ⋅dA

dΦ Maxwell defined a fictitious “displacement current”: I = ε E D 0 dt

Thus, a changing Electric Field creates a Magnetic Field !

The displacement current

ε A Q = CV = 0 (Ed) = ε EA = ε Φ d 0 0 E dΦ dQ I = ε E = = I D 0 dt dt C

dΦ Between the plates: I = ε E ; I = 0 D 0 dt C Outside the capacitor: I D = 0; IC ≠ 0

The B-field in between the plates is exactly the same as the field created by the current outside. The fictitious “displacement current” gives an idea of continuity of the current, even though there is no charge flowing in the space separating the plates Maxwell’s equations

r r Q ∫ E ⋅dA = encl Gauss’ Law for E-fields Surf ε 0 r r ∫ B ⋅dA = 0 Gauss’ Law for B-fields Surf r r ⎛ d ⎞ ⎜ Φ E ⎟ Ampere’s Law ∫ B ⋅dl = µ0 Ithru + ε 0 loop ⎝ dt ⎠ r r dΦ ∫ E ⋅dl = − B Faraday’s Law loop dt