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Project PHYSNET Physics Bldg. Michigan State University East Lansing, MI

MISN-0-142

MAGNETIC INDUCTION

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1

MAGNETIC INDUCTION

by

J. S. Kovacs and P. SignellMichigan State University

1. Introductiona . O v e r v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

b. Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1c. The Flux Integrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2d. The Direction of Positive Flux ...........................2e. Induced Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3f. Induced Voltage and Flux Change Rate . . . . . . . . . . . . . . . . . . 3g. The Power Transferred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Example of Lorentz-Force Inductiona. A Loop Enters A Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 4b. The Loop Travels Through the Magnetic Field . . . . . . . . . . 5c. The Loop Leaves the Magnetic Field .....................5d. The Time Rate of Change of the Flux . . . . . . . . . . . . . . . . . . . 6

3. Induction by Time-Changing Ba. The Faraday-Henry Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6b. Example: Linearly Increasing B . . . . . . . . . . . . . . . . . . . . . . . . . 6

4. Lenzs Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5. Self-Inductance, Inductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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MISN-0-142 5

-2l

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B vl

-B vl

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Figure 5. The induced potential difference around the loopof Fig. 4 as a function of the position of the loop.

we get for the induced potential of point b with respect to point a:

Vind,ba = v B . (4)

Thus any induced current will flow from a to b (from higher potential tolower, in the direction of the Lorentz Force).

2b. The Loop Travels Through the Magnetic Field. As the looptravels across the magnetic field the induced potential difference alongthe top and bottom arms of the loop will be zero as is readily seen fromthe vector form of the Lorentz force. The potential difference between theends of the trailing edge of the loop will be the same as for the leadingedge:

Vind,cd = v B . (5)

As we make a complete circuit around the loop the induced potential

differences along the two ends oppose each other and the net potentialdifference is zero:Vind,loop = 0 . (6)

2c. The Loop Leaves the Magnetic Field. When the leading edgeof the coil reaches the far edge of the field and emerges into a regionof zero magnetic field, its induced potential drops to zero and the looppotential difference is:

Vind,dc = + v B , (7)

which would cause current to flow around the loop in a clockwise direction.Fig. 5 shows a plot of the loop potential difference, Vdcba, vs. x.

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MISN-0-142 6

2d. The Time Rate of Change of the Flux. The induced potentialdifference is simply related to the time rate of change of the magnetic fluxenclosed by the loop:

Vind,dcba =

. (8)This comes about because the magnetic field is constant so the flux in-tegral, Eq. (1), is just = B dA/dt where A is the area of the loopcontaining non-zero B. That area is A = r so dA/dt = dr/dt = v.Then = B v and Eq. (8) is proved. We will find that Eq. (8) is notrestricted to Lorentz force cases but is also true where the loop is station-ary (not moving) in the field but the magnetic field itself is increasing ordecreasing with time.

3. Induction by Time-Changing B

3a. The Faraday-Henry Law. Time-changing magnetic fields pro-duce induced electric fields, hence potential differences, in a manner de-scribed by the so-called Faraday-Henry law. This law relates the lineintegral of the induced electric field to the time-derivative of the surface-integrated magnetic field:

LS

Eind d

=

d

dtSL

B d

S . (9)

Now we can substitute (Vind) for the left side and () for the rightside to get the simpler-appearing form:

Vind,LS = SL . (10)

Note that this equation is exactly the same as the equation for Lorentz-force induction, Eq. (8).

3b. Example: Linearly Increasing B. Suppose a coil having Nturns and area A is placed perpendicular to a magnetic field that is in-creasing linearly with time:

B = B0 + B1 t ,

where B0 and B1 are constants. The induced voltage is, by Eq. (10):

Vind = = N A B1 .

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MISN-0-142 9

induced potential difference: the total distributed potential differ-ence encountered in traversing a circuit due to motion of the circuit ina magnetic field or due to the circuit being in a time-changing magnetic

field. induced voltage: just another name for induced potential differ-

ence (which see).

self-inductance: the negative of the induced voltage around a loopdivided by the time-rate-of-change of magnetic flux through any surfacebounded by that loop: L = Vinduced/. The minus sign shows thatthe induced voltage opposes the change in the flux. The SI unit ofinductance is the henry, abbreviated H: H V s A1.

inductor: a circuit element whose purpose is to provide self-inductance, the electrical circuit analog of mechanical inertia (mass).An inductor is usually in the shape of a solenoid or a toroid. The self-inductance of an inductor depends on the geometry of the inductor,and the magnetic susceptibility of the materials of which the inductoris constructed. Inductors in electronic circuits typically are in the mHrange.

Lenzs law: a law stating that induced phenomena oppose the induc-

ing phenomena. For example, an induced current produces a magneticfield which opposes the changes in the magnetic field that caused thecurrent to be induced.

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MISN-0-142 PS-1

PROBLEM SUPPLEMENT

Note: Problems 1-3 also appear in this modules Model Exam.

1. A circular loop, of radius R = 0.20 meters, in the plane of the pageis in a region where a magnetic field B is directed into the page (see

sketch). The magnitude of B varies with time according to:

| B| = B0 At2,

where B0 = 2.0 T (teslas) and A = 0.30T/s2

are constants.x x

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B`

directed into page

a. Find the magnitude and direction of the induced potential differ-ence, Vind, around this loop at the instant t = 0.20sec. [F]

Is Vind constant with time? [A]

Check that the dimensions of Vind follow correctly from the dimen-sions of what goes into your calculations. [H]

b. Find the magnitude and direction of the tangential component ofthe electric field vector at any point on this loop. [J]

c. What can the Faraday-Henry Law tell you about the normal (radial)

component of E at any point on the loop? [E]

What does symmetry tell you? [I]

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MISN-0-142 PS-2

2.

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S T

P

Q

A A'

This same loop is placed in a region of uniform field B, | B| = 2.0 T,

directed into the page. The loop is rotated about the axis AA

so thatpoint P comes out of the page, point Q goes into the page. In 0.02secthe loop is flipped over so that P is where Q was and vice versa.

a. What is the average Vind induced around this loop? [B]

b. What is the direction of this Vind? (Would it drive the inducedcurrent in the direction SP T or T P S? [D]

3. As shown in the figure, lines of magnetic field are directed perpendic-ular to a circular loop of wire of radius r.

a. If the magnitude of the field is given by B = B0et/, where B0 and

are constants, what is the Vind induced in the loop as a functionof time? [G] (In the figure, ds denotes the positive direction aroundthe loop.)

B

r

ds

b. Answer the same question if the loop is titled through 60.0, [L] or90.0. [K]

4. Using the definitions of the tesla (p. 359) and the volt, in terms offundamental dimensions, show that both sides of the Faraday-HenryLaw have the same units. [C]

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MISN-0-142 PS-3

5.

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A square loop of wire is stationary within a time-dependent magneticfield that is confined to a square region of space. The length of one sideof the loop is and the loop extends /2 into the field (see sketch). Themagnetic field varies with time as B = B0sin(t). Use these values toanswer the questions below: B0 = 1.0 T, = 2.0/ s, R = 1.0 and = 1.0 m.

a. At t = 3.5s what is Vind? [M]

b. At that same time what is the power dissipation through the resis-tor? [O] Help: [S-3]

c. What is the work done by the field from t = 0.0 s to t = 3.5s? [N]Help: [S-2]

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MISN-0-142 PS-4

Brief Answers:

A. No.

B. 25 volts. Help: [S-1]

C. See text.

D. Again, use Lenzs Law to find that the direction of decreasing Vind,the downhill direction for the current, the direction the current willmove, is SP T.

E. Nothing, in the case of this circular path.

F. 0.015 volts, clockwise.

G.r2B

.

Lenzs Law verifies, direction same as ds.

Dimensions: m2 T s1 = kgm2 s2 C1 = N C1 = volt

H. See definition of the dimension of tesla, p. 359 of AF.

I. Symmetry tells you that whatever En (the component of E normal topage) is, it is the same at every point on the circle. Because the changein B is entirely perpendicular to the page, En in fact is zero. (Referto Lenzs Law in this modules text).

J. Et = 0.012 volt/m = 0.012 N/C.

K. Zero.

L.

r2B

2 .

M. Vind = 2B0

2cos(t) = V

N. W =1

R

2

2B0

2 t

2+

1

4sin(2t)

= 1.752 J Help: [S-3]

O. P = V2ind/R =2 V2

1.0 = 2 A V = 2 W Help: [S-2]

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MISN-0-142 AS-1

SPECIAL ASSISTANCE SUPPLEMENT

S-1 (from PS, Problem 2a)

The average value of a sine function over a half cycle is:

sin x =

0

sin x dx0

dx=

| cos x|0

= . . .

S-2 (from PS, Problem 5b)

The power-voltage-resistance relation is referenced in this modules IDSheet.

S-3 (from PS, Problem 5c)

The energy-power-time relation is referenced in this modules ID Sheet.

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MISN-0-142 ME-1

MODEL EXAM

1. See Output Skills K1-K2 in this modules ID Sheet. The actual exammay have one or more of these skills, or none.

2. A circular loop, of radius R = 0.20 meters, in the plane of the pageis in a region where a magnetic field B is directed into the page (see

sketch). The magnitude of B varies with time according to:

| B| = B0 At2,

where B0 = 2.0 T (teslas) and A = 0.30T/s2

are constants.x x

x

x

x

x

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x

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x

x

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x

B directed into page

a. Find the magnitude and direction of the induced potential differ-ence, Vind, around this loop at the instant t = 0.20 sec.

Is Vind constant with time?

Check that the dimensions of Vind follow correctly from the dimen-sions of what goes into your calculations.

b. Find the magnitude and direction of the tangential component ofthe electric field vector at any point on this loop.

c. What can the Faraday-Henry Law tell you about the normal (radial)

component of E at any point on the loop?

What does symmetry tell you?

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MISN-0-142 ME-2

3.

xx x

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x x x x x

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S T

P

Q

A A'

This same loop is placed in a region of uniform field B, | B| = 2.0 T,

directed into the page. The loop is rotated about the axis AA

so thatpoint P comes out of the page, point Q goes into the page. In 0.02 secthe loop is flipped over so that P is where Q was and vice versa.

a. What is the average Vind induced around this loop?

b. What is the direction of this Vind? (Would it drive the inducedcurrent in the direction SP T or T P S?

4. As shown in the figure, lines of magnetic field are directed perpendic-ular to a circular loop of wire of radius r.

a. If the magnitude of the field is given by B = B0et/, where B0and are constants, what is the Vind induced in the loop as afunction of time? (In the figure, ds denotes the positive directionaround the loop.)

B

r

ds

b. Answer the same question if the loop is titled through 60.0, or90.0.

Brief Answers:

1. See this moduless text.

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MISN-0-142 ME-3

2. See this modules Problem Supplement, problem 1.

3. See this modules Problem Supplement, problem 2.

4. See this modules Problem Supplement, problem 3.

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