physics- magnetic field

Magnetic Forces & Fields

Unless otherwise noted, all content on this page is © Cengage Learning.

Problems 967

69. A small, circular washer of radius a 5 0.500 cm is held directly below a long, straight wire carrying a current of I 5 10.0 A. The washer is located h 5 0.500 m above the top of a table (Fig. P31.69). Assume the magnetic field is nearly constant over the area of the washer and equal to the magnetic field at the center of the washer. (a) If the washer is dropped from rest, what is the mag-nitude of the average induced emf in the washer over the time interval between its release and the moment it hits the tabletop? (b) What is the direction of the induced current in the washer?

h

I

Figure P31.69

70. Figure P31.70 shows a compact, circular coil with 220 turns and radius 12.0 cm immersed in a uniform magnetic field parallel to the axis of the coil. The rate of change of the field has the constant magnitude 20.0 mT/s. (a) What additional information is neces-sary to determine whether the coil is carrying clock-wise or counterclockwise current? (b) The coil overheats if more than 160 W of power is delivered to it. What resistance would the coil have at this critical point? (c) To run cooler, should it have lower resis-tance or higher resistance?

71. A rectangular coil of 60 turns, dimensions 0.100 m by 0.200 m, and total resistance 10.0 V rotates with angu-lar speed 30.0 rad/s about the y axis in a region where a 1.00-T magnetic field is directed along the x axis. The time t 5 0 is chosen to be at an instant when the plane of the coil is perpendicular to the direction of B

S.

Calculate (a) the maximum induced emf in the coil, (b) the maximum rate of change of magnetic flux through the coil, (c) the induced emf at t 5 0.050 0 s, and (d) the torque exerted by the magnetic field on the coil at the instant when the emf is a maximum.

72. Review. In Figure P31.72, a uniform magnetic field decreases at a constant rate dB/dt 5 2K, where K is a positive constant. A circular loop of wire of radius a containing a resistance R and a capacitance C is placed

AMT

Q/C

Q/CS

The wheels make good electrical contact with the rails, so the axle, rails, and R form a closed-loop cir-cuit. The only significant resistance in the circuit is R. A uniform magnetic field B 5 0.080 0 T is vertically downward. (a) Find the induced current I in the resis-tor. (b) What horizontal force F is required to keep the axle rolling at constant speed? (c) Which end of the resistor, a or b, is at the higher electric poten-tial? (d) What If? After the axle rolls past the resistor, does the current in R reverse direction? Explain your answer.

67. Figure P31.67 shows a stationary conductor whose shape is similar to the letter e. The radius of its circu-lar portion is a 5 50.0 cm. It is placed in a constant magnetic field of 0.500 T directed out of the page. A straight conducting rod, 50.0 cm long, is pivoted about point O and rotates with a constant angular speed of 2.00 rad/s. (a) Determine the induced emf in the loop POQ. Note that the area of the loop is ua2/2. (b) If all the conducting material has a resistance per length of 5.00 V/m, what is the induced current in the loop POQ at the instant 0.250 s after point P passes point Q?

This rod rotatesabout O.

P

QO

out

!a

BS

Figure P31.67

68. A conducting rod moves with a constant velocity in a direction perpendicular to a long, straight wire carry-ing a current I as shown in Figure P31.68. Show that the magnitude of the emf generated between the ends of the rod is 0e 0 5

m0vI,

2pr

In this case, note that the emf decreases with increas-ing r as you might expect.

S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70


968 Chapter 31 Faraday’s Law

rent I in the plane of the loop (Fig. P31.76). The total resistance of the loop is R. Derive an expression that gives the current in the loop at the instant the near side is a distance r from the wire.

77. A long, straight wire carries a current given by I 5 I max sin (vt 1 f). The wire lies in the plane of a rectan-gular coil of N turns of wire as shown in Figure P31.77. The quantities Imax, v, and f are all constants. Assume I max 5 50.0 A, v 5 200p s21, N 5 100, h 5 w 5 5.00 cm, and L 5 20.0 cm. Determine the emf induced in the coil by the magnetic field created by the current in the straight wire.

w

h

L

I

Figure P31.77

78. A thin wire , 5 30.0 cm long is held parallel to and d 5 80.0 cm above a long, thin wire carrying I 5 200 A and fixed in position (Fig. P31.78). The 30.0-cm wire is released at the instant t 5 0 and falls, remaining parallel to the current-carrying wire as it falls. Assume the fall-ing wire accelerates at 9.80 m/s2. (a) Derive an equation for the emf induced in it as a function of time. (b) What is the minimum value of the emf? (c) What is the maximum value? (d) What is the induced emf 0.300 s after the wire is released?

Challenge Problems

79. Two infinitely long solenoids (seen in cross section) pass through a circuit as shown in Figure P31.79. The magnitude of B

S inside each is the same and is increas-

ing at the rate of 100 T/s. What is the current in each resistor?

BoutSBin

S

0.500 m 0.500 m

0.500 m6.00 ! 5.00 !3.00 !

r2 = 0.150 mr1 = 0.100 m

Figure P31.79

80. An induction furnace uses electromagnetic induction to produce eddy currents in a conductor, thereby rais-ing the conductor’s temperature. Commercial units

M

d

,

I

Figure P31.78

S

S

with its plane normal to the field. (a) Find the charge Q on the capacitor when it is fully charged. (b) Which plate, upper or lower, is at the higher potential? (c) Dis-cuss the force that causes the separation of charges.

73. An N -turn square coil with side , and resistance R is pulled to the right at constant speed v in the presence of a uniform magnetic field B acting perpendicular to the coil as shown in Figure P31.73. At t 5 0, the right side of the coil has just departed the right edge of the field. At time t, the left side of the coil enters the region where B 5 0. In terms of the quantities N, B, ,, v, and R, find symbolic expressions for (a) the magnitude of the induced emf in the loop during the time interval from t 5 0 to t, (b) the magnitude of the induced cur-rent in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the field. (e) What is the direction of the induced current in the loop? (f) What is the direction of the magnetic force on the loop while it is being pulled out of the field?

FappS

B " 0S

BinS

Figure P31.73

74. A conducting rod of length , moves with velocity vS parallel to a long wire carrying a steady current I. The axis of the rod is maintained perpendicular to the wire with the near end a distance r away (Fig. P31.74). Show that the magnitude of the emf induced in the rod is0e 0 5

m0Iv2p

ln a1 1,

rb

r

I

!

vS

Figure P31.74

75. The magnetic flux through a metal ring varies with time t according to FB 5 at 3 2 bt 2, where FB is in webers, a 5 6.00 s23, b 5 18.0 s22, and t is in seconds. The resistance of the ring is 3.00 V. For the interval from t 5 0 to t 5 2.00 s, deter-mine the maximum current induced in the ring.

76. A rectangular loop of dimen-sions , and w moves with a constant velocity vS away from a long wire that carries a cur-

S

S

M

S

IR

r w

!vS

Figure P31.76


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70





w

h

L

I

Figure P31.77


Challenge Problems




BoutSBin

S

0.500 m 0.500 m

0.500 m6.00 ! 5.00 !3.00 !

r2 = 0.150 mr1 = 0.100 m

Figure P31.79


M

d

,

I

Figure P31.78

S

S



FappS

B " 0S

BinS

Figure P31.73


m0Iv2p

ln a1 1,

rb

r

I

!

vS

Figure P31.74



S

S

M

S

IR

r w

!vS

Figure P31.76


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70

given:dB/dt = -Kradius a

resistance Rcapacitance C

find:(a) Q

to find Q:Q = Cℰ

ℰ = -d/dt(NBA)find ℰ:

what is N?N = 1

find A: A = πa2

plug in:ℰ = (-1)(dB/dt)(πa2)Q = C(-1)(dB/dt)(πa2)

Q = Cπa2Kinclude given values:


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70





w

h

L

I

Figure P31.77


Challenge Problems




BoutSBin

S

0.500 m 0.500 m

0.500 m6.00 ! 5.00 !3.00 !

r2 = 0.150 mr1 = 0.100 m

Figure P31.79


M

d

,

I

Figure P31.78

S

S



FappS

B " 0S

BinS

Figure P31.73


m0Iv2p

ln a1 1,

rb

r

I

!

vS

Figure P31.74



S

S

M

S

IR

r w

!vS

Figure P31.76


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70

part (b)is there an

increasing or decreasing Φ?yes,

decreasing (into the screen)

what direction would a new

B field point to oppose this

change in Φ?

into the screen

what direction must current flow through the loop to produce a B field in this direction?

clockwisewhich capacitor plate will accumulate positive charge (higher potential)?

the upper plate


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70





w

h

L

I

Figure P31.77


Challenge Problems




BoutSBin

S

0.500 m 0.500 m

0.500 m6.00 ! 5.00 !3.00 !

r2 = 0.150 mr1 = 0.100 m

Figure P31.79


M

d

,

I

Figure P31.78

S

S



FappS

B " 0S

BinS

Figure P31.73


m0Iv2p

ln a1 1,

rb

r

I

!

vS

Figure P31.74



S

S

M

S

IR

r w

!vS

Figure P31.76


Problems 967


h

I

Figure P31.69



S.



AMT

Q/C

Q/CS




P

QO

out

!a

BS

Figure P31.67


m0vI,

2pr


S

R C

BinS

Figure P31.72

r

I

!vS

Figure P31.68

BS

Figure P31.70

part (c)

changing B flux through loop induces clockwise E field in loop

electric force acts on charges in wire

∫[(dy)/(x2 + y2)3/2] = y/[x2(x2 + y2)1/2] | = 1/x2


926 Chapter 30 Sources of the Magnetic Field

direction of the field produced at P if the current is 3.00 A?

14. One long wire carries current 30.0 A to the left along the x axis. A second long wire carries current 50.0 A to the right along the line (y 5 0.280 m, z 5 0). (a) Where in the plane of the two wires is the total magnetic field equal to zero? (b) A particle with a charge of 22.00 mC is moving with a velocity of 150 i Mm/s along the line (y 5 0.100 m, z 5 0). Calculate the vector magnetic force acting on the particle. (c) What If? A uni-form electric field is applied to allow this particle to pass through this region undeflected. Calculate the required vector electric field.

15. Three long, parallel conductors each carry a current of I 5 2.00 A. Figure P30.15 is an end view of the conduc-tors, with each current coming out of the page. Taking a 5 1.00 cm, determine the magnitude and direction of the magnetic field at (a) point A, (b) point B, and (c) point C.

I

I

aa

a

a

aBA C I

Figure P30.15

16. In a long, straight, vertical lightning stroke, electrons move downward and positive ions move upward and constitute a current of magnitude 20.0 kA. At a loca-tion 50.0 m east of the middle of the stroke, a free elec-tron drifts through the air toward the west with a speed of 300 m/s. (a) Make a sketch showing the various vec-tors involved. Ignore the effect of the Earth’s magnetic field. (b) Find the vector force the lightning stroke exerts on the electron. (c) Find the radius of the elec-tron’s path. (d) Is it a good approximation to model the electron as moving in a uniform field? Explain your answer. (e) If it does not collide with any obstacles, how many revolutions will the electron complete during the 60.0-ms duration of the lightning stroke?

17. Determine the magnetic field (in terms of I, a, and d) at the origin due to the current loop in Figure P30.17. The loop extends to infinity above the figure.

P

I

Iu

Figure P30.13

AMTM

Q/C

S

rent I 2. The total magnetic field at the origin due to the current-carrying wires has the magnitude 2m0I1/(2pa). The current I 2 can have either of two pos-sible values. (a) Find the value of I 2 with the smaller magnitude, stating it in terms of I 1 and giving its direc-tion. (b) Find the other possible value of I 2.

x

I2 I1

2a–2a 0

Figure P30.9

10. An infinitely long wire carrying a current I is bent at a right angle as shown in Figure P30.10. Determine the magnetic field at point P, located a distance x from the corner of the wire.

x

P

I

I

Figure P30.10

11. A long, straight wire carries a current I. A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius r as shown in Figure P30.11. Determine the magnetic field at point P, the center of the arc.

r

PI

Figure P30.11

12. Consider a flat, circular current loop of radius R car-rying a current I. Choose the x axis to be along the axis of the loop, with the origin at the loop’s center. Plot a graph of the ratio of the magnitude of the mag-netic field at coordinate x to that at the origin for x 5 0 to x 5 5R. It may be helpful to use a programmable calculator or a computer to solve this problem.

13. A current path shaped as shown in Figure P30.13 pro-duces a magnetic field at P, the center of the arc. If the arc subtends an angle of u 5 30.08 and the radius of the arc is 0.600 m, what are the magnitude and

S

S

4. In 1962, measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the magnitude of the tornado’s field was B 5 1.50 3 1028 T pointing north when the tornado was 9.00 km east of the observatory, what cur-rent was carried up or down the funnel of the tornado? Model the vortex as a long, straight wire carrying a current.

5. (a) A conducting loop in the shape of a square of edge length , 5 0.400 m carries a current I 5 10.0 A as shown in Figure P30.5. Calculate the magnitude and direction of the magnetic field at the center of the square. (b) What If? If this conductor is reshaped to form a circular loop and carries the same current, what is the value of the magnetic field at the center?

I

!

Figure P30.5

6. In Niels Bohr’s 1913 model of the hydrogen atom, an electron circles the proton at a distance of 5.29 3 10211 m with a speed of 2.19 3 106 m/s. Compute the magnitude of the magnetic field this motion produces at the location of the proton.

7. A conductor consists of a circular loop of radius R 5 15.0 cm and two long, straight sections as shown in Fig-ure P30.7. The wire lies in the plane of the paper and carries a current I 5 1.00 A. Find the magnetic field at the center of the loop.

RI

Figure P30.7 Problems 7 and 8.

8. A conductor consists of a circular loop of radius R and two long, straight sections as shown in Figure P30.7. The wire lies in the plane of the paper and carries a current I. (a) What is the direction of the magnetic field at the center of the loop? (b) Find an expression for the magnitude of the magnetic field at the center of the loop.

9. Two long, straight, parallel wires carry currents that are directed perpendicular to the page as shown in Figure P30.9. Wire 1 carries a current I1 into the page (in the negative z direction) and passes through the x axis at x 5 1a. Wire 2 passes through the x axis at x 5 22a and carries an unknown cur-

M

W

S

S

given:Ix

find:B

to find B:B = (μ0/4π)∫[(Ids ╳ ȓ)/r2]

(ds ╳ ȓ) = 0for horizontal wire:

for vertical wire:

+y(ĵ)

+x(î)

y = -∞

wire segment ds(0,y)

for vertical wire:ds = ĵdy

(x,0)

r = (x,0) - (0,y) = xî - yĵ

r

r = √(x2 + y2)

to find ds ╳ ȓ:ds ╳ ȓ = [(ĵdy) ╳ (xî - yĵ)]/r

(ȓ = r/r)ds ╳ ȓ = (-kxdy)/r^

B = -k(μ0/4π)∫[(Ixdy)/r3]plug in: ^-∞

0

B = -k(μ0Ix/4π)∫[(dy)/(x2 + y2)3/2]pull out constants:

^-∞

0

-∞

0

-∞

0

plug in: B = -k(μ0I/4πx)^

checkdirectionwith RHR