## Magnetic Sources

Homework Solutions

Questions 1, 2, 3, 4, 9, 18, 19

Problems 30.8A, 22, 23, 25, 32, 40, 60, 64

Q1Is the magnetic field created by a current loop uniform?Not at all; it varies with position -- it varies in direction and magnitude -- as shown in this sketch:

Q2A current in a conductor produces a magnetic field that can be calculated using the Biot-Savart law. Since current is defined as the rate of flow of charge, what can you conclude about the magentic field produced by a stationary charge? What about moving charges?The magnetic field produced by a

stationarycharge iszero.A moving chargeisa current so itwill producea magnetic field.

Q3Two parallel wires carry currents in opposite directions. Describe the nature of the resultant magnetic field created by the two wires at points

(a) between the wires and

or

Betweenthe wires, the magnetic fields from the two wires point in thesamedirection so the resulting or total magnetic field will belarge.(b) outside the wires in a plane containing them.

or

Outsidethe wires, the two magnetic fields from the two currents are inoppositedirections so they will tend to cancel each other and the resulting or total magnetic field will besmall.

Q4Explain why two parallel wires carrying currents inoppositedirections repel each other.You may want to look at the diagrams for

Question 3. In particular, consider this diagram,

or

The magnetic field of the left-hand current causes a force on the right-hand current that is a repulsion so we know -- from Newton's Third Law -- that the

twocurrentsrepel each other.

Q9Describe the similarities between Ampere's law in magnetism and Gauss's law in electrostatics.Both associate an integral around a closed surface or line with something that goes through the surface or line of the integral. Gauss's Law says the electric flux through a closed surface is proportional to the charge within that surface.. Ampere's Law says the line integral of

B-dsaround a closed path is proportional to the current that passes through that path.

Q18Will a nail be attracted to either pole of a magnet? Explain what is happening inside the nail when placed near a magnet.Yes, a nail

will be attractedtoeither poleof a magnet. The magnet causes the boundaries between magnetic domains to move slightly, giving a net magnetic field to the nail.path If the nail is near the othermagnetic pole then different magnetic domain boundaries move.

Q19The north-seeking ppole of a magnet is attracted toward the geographic North Pole of Earth. Yet, like poles reple. What is going on here?The magnetic pole near Earth's geographic North Pole is what would otherwise be called a

south-seeking magnetic pole.

30.8A,A conductor consists of a circular loop of radius R and two straight, long sections, as in Figure P30.8. The wire lies in the plane of the paper and carries a current I. Determine the magnitude and direction of the magnetic field at the center of the loop.We know how to handle the magnetic field due to a

straight wire;At a distance R from the wire, the magnetic field is

And we know how to handle the magnetic field at the center of a

circular loop;At the center of the wire, the magnetic field is

So our total magnetic field at the center of the circular loop and distance R from the long, straight wire, is the

sumof these,

30.22,A closely wound, long solenoid of overall length 30.0 cm (0.30 m) has a magnetic field of 4.00 x 10 - 4 T at its center produced by a current of 1.00 A through its windings. How many turns of wire are on the solenoid?The magnetic field inside a solenoid is given by

or

n = B / I n = (4 x 10

^{ - 4}) / [(4 x 10^{ - 7})(1)]n = 318 turns/m

n = N/L

N = n L

N = (318 turns/m)(0.30 m)

N = 95 turns

30.23,A superconducting solenoid is to generate a magnetic field of 10.0 T.(a) If the solenoid windings has 2000 turns/m, what is the required current?

or

I = B / n I = (10)/[(4 x 10

^{ - 7})(2000)]I = 3979 A

That is an enormous current; but then that is an enormous magnetic field!

30.25,Some superconducting alloys at very low temperatres can carry very high currents. For example Nb_{3}Sn wire at 10 K can carry 1000 A and maintain its superconductivity. Determine the maximum value of B achievable in a solenoid of length 25 cm (0.25 m) if 1000 turns of Nb_{3}Sn wire are wrapped on the outside surface.B = (4 x 10

^{ - 7})(1000/0.25)(1000)B = 5 T

30.32,In Figure P30.32, both currents are in the negative x direction.(a) Sketch the magnetic field pattern in the yz-plane.

Remember, this is a

roughsketch!

30.40,Consider the hemispherical closed surface in Figure P30.40. If the hemisphere is in a uniform magnetic field that makes and angle theta with the vertical, calculate the magnetic flux through(a) the flat surface S

_{1}and= A B cos A = R

^{2}= R

^{2}B cos(b) the hemispherical surface S

_{2}.According to Gauss's Law for

magnetic flux, thetotalmagnetic flux iszero. Since we know the flux comingintothe surface ( = R^{2}B cos ) that (or thenegative) of that is also the flux comingout ofthe surface -- the flux through S_{2},= - R ^{2}B cos

30.60,A lightning bolt may carry a current of 10,000 A for a short period of time. What is the resulting magnetic field 100 m from the botl?B = [(4 x 10

^{ - 7})(10 000)]/[(2 ) (100)]B = 2 x 10

^{ - 5}T

30.64Two parallel conductors carry current in opposite directions as showin in Figure P30.64. One conductor carries a current of 10A. Point A is at the midpoint between the wires and point C is a distance d/2 to the right of the 10 A current. If d = 18 cm = 0.18 m and I is adusted so that the magnetic field at C is zero, find(a) the value of the current I and

(b) the value of the magnetic field at A.

For Point

C,B _{right}= [(4 x 10^{ - 7})(10)]/[(2 )(0.09)] = 2.22 x 10^{ - 5}TB

_{left}= [(4 x 10^{ - 7})(I)]/[(2 )(0.27)] = 2.22 x 10^{ - 5}TI = (2.22 x 10

^{ - 5}) (0.27) / 2 x 10^{ - 7}I = 30 A

For Point

A,B _{right}= [(4 x 10^{ - 7})(10)]/[(2 )(0.09)] = 2.22 x 10^{ - 5}TB

_{left}= [(4 x 10^{ - 7 })( 30 )]/[(2 )(0.09)] = 6.66 x 10^{ - 5}TUsing the Right Hand Rule, we see that these two fields point in the same direction -- "up" or "out of the page" at point A

B _{total}= B_{right}+ B_{left}= 8.88 x 10^{ - 5}T(c) Doug Davis, 2002; all rights reserved

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