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Chapter 29--Examples

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Problem

a) Derive the equation relating the total charge Q that flows through a search coil (Conceptual Example 29.3 in Section 29.2) to the magnetic field B. The search coil has N turns, each with area A and the flux through the coil is decreased from its initial value to zero in a time t. The resistance of the coil is R and the total charge is Q=I t where I is the average current induced by the change in flux

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From Example 29.3

A search coil is a practical way to measure magnetic field strength. It uses a small, closely wound coil with N turns. The coil, of area A, is initially held so that its area vector A is aligned with a magnetic field with magnitude B. The coil is quickly pulled out of the field or rotated.

Initially the flux through the field is =NBA. When it leaves the field or rotated, goes to zero. As decreases, there is a momentary induced current which is measured. The amount of current is proportional to the field strength.

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Solution

EMF=-/t–

Where =NBA and =0

EMF=NBA/tV=iR where V=EMF

EMF=NBA/t=iR i=NBA/Rt

Q=it=NBA/R

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Part B

In a credit card reader, the magnetic strip on the back of the card is “swiped” past a coil within the reader. Explain using the ideas of the search coil how the reader can decode information stored in the pattern of magnetization in the strip.

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Solution

The card reader contains a search coil.The search coil produces high EMF and

low EMF as the card is swiped.A high EMF is treated as a binary 1 and

a low EMF is treated as a binary 0Ascii information (character codes

between 1 and 64) can be stored there in a few bits (6-bits or 26).

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Problem

A circular loop of flexible iron wire has an initial circumference of 165 cm but its circumference is decreasing at a rate of 12 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with a magnitude of 0.5 T

a) Find the EMF induced in the loop at the instant when 9 s have passed.

b) Find the direction of the current in the loop as viewed looking in the direction of the magnetic field.

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EMF=-/t=-B*A/t

=BA d/dt=BdA/dt c=2r so A=r2

r=c/2 so A=c2/4 dA/dt=d/dt(c2/4c/4)dc/dt d/dt= Bc/2)dc/dt

Where B=0.5 dc/dt=0.12 m/s At 9s, c=1.65-.12*(9s)=0.57 m

d/dt=5.44 x 10-3 V Since is decreasing, the EMF is positive. Since

EMF positive, point thumb along A and look at fingers. They curl counterclockwise.

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Problem

Suppose the loop in the figure below is a) Rotated about the y-axisb) Rotated about the x-axisc) Rotated about an edge parallel to the z-axis.What is the maximum induced EMF in each case if A=600 cm2, w=35

rad/s, and B=.45T?

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Case a—Rotating about y-axis

VBAEMF

tBAtBAdt

dtbut

BAdt

dAB

dt

d

dt

dEMF

945.035*06.0*45.0

sincos

cos

max

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Case b—rotating about the x-axis

The normal to the surface is in the same direction as B during this rotation. Thus BA=constant

d/dt(constant)=0 so EMF=0

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Case c—Rotating about z-axis

VBAEMF

tBAtBAdt

dtbut

BAdt

dAB

dt

d

dt

dEMF

945.035*06.0*45.0

sincos

cos

max

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Problem

The figure below shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x>>R. Consequently, the magnetic field due to current i in the larger loop is nearly constant throughout the smaller loop. Suppose that x is increasing at a constant rate, v.

1) Determine the magnetic flux though area of the smaller loop as a function of x.

2) In the smaller loop find1) The induced EMF2) The direction of the induced current.

x

Radius=r

Radius=R

i

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Magnetic Field of a Circular Loop

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20

23

22

20

22 x

iR

Rx

iRB

By the RH Rule, the field is upward in the small ring

23

20

23

20

2

2

rx

iRAB

rAandx

iRB

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EMF in smaller loop

42

20

43

32

202

3

20

23

20

3

2

31

1

22

2

xvr

iREMF

vdt

dxbut

dt

dx

xxdt

d

xdt

dr

iRr

x

iR

dt

d

dt

dEMF

rx

iRAB

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Direction of current

Assume that normal to smaller loop is positive upwards

B, then, is in positive directionBut is decreasing or has a negative d

d /dtA negative * negative=positive so EMF is

+RH Thumb in up direction, fingers curl

counter clockwise!