Mid-exam2

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Fig. 3

Fig. 1

Fig. 2 TAIYUAN UNIVERSITY OF SCIENCE AND TECHNOLOGY

The College of Applied Science

Department of Physics

Engineering Physics November 2014

The Exam of Midsemester

(Including Chapter 19-27)

Please write your name and student number on the top right corner of the first page, and write down your solutions on the A4 papers .

Problem 1 (20 points)

(a ) Consider a uniformly charged thin-walled right circular cylindrical

shell having total charge Q , radius R , and height h . Determine the

electric field and the electric potential at a point a distance d from the

right side of the cylinder as shown in Fig. 1. (b ) What If? Consider

now a solid cylinder with the same dimensions and carrying the same

charge, uniformly distributed through its volume.

Problem 2 (10 points)

A solid, insulating sphere of radius a has a total charge Q uniformly

distributed over its surface . Concentric with this sphere is an uncharged,

conducting, hollow sphere whose inner and outer radii are b and c as

shown in Fig. 2. (a ) Find the magnitude of the electric field in the

regions r < a , a < r c .(b ) Determine the induced

charge per unit area on the inner and outer surfaces of the hollow sphere.

Problem 3 (10 points)

An infinitely long insulating cylinder of radius R has a volume charge density that varies with the

radius as

Where ρ0, a, and b are positive constants and r is the distance from the axis of the cylinder. Use Gauss’s law to determine the magnitude of the electric field at radial distances (a) r R.

Problem 4 (10 points)

A slab of insulating material has a nonuniform positive charge density ρ

=Cx 2, where x is measured from the center of the slab as shown in Fig. 3

and C is a constant. The slab is infinite in the y and z directions. Derive

expressions for the electric field in (a) the exterior regions and (b) the

interior region of the slab (-d /2

2 Fig. 4

Fig. 5

Fig. 6 Fig. 7

Problem 5 (10 points)

Consider two long, parallel, and oppositely charged wires of radius r with their centers separated by a distance D that is much larger than r . Assuming the charge is distributed uniformly on the

surface of each wire, show that the capacitance per unit length of this pair of wires is

Problem 6 (10 points)

A long, cylindrical conductor of radius R carries a current I as shown

in Fig. 4. The current density J , however, is not uniform over the

cross section of the conductor but is a function of the radius

according to J = br , where b is a constant. Find an expression for the

magnetic field magnitude B (a ) at a distance r 1

distance r 2> R , measured from the axis.

Problem 7 (10 points)

A thin copper bar of length l is supported horizontally by two

(nonmagnetic) contacts. The bar carries current I 1 in the x direction

as shown in Fig. 5. At a distance h below one end of the bar, a long,

straight wire carries a current I 2 in the z direction. Determine the

magnetic force exerted on the bar.

Problem 8 (10 points)

A loop of wire in the shape of a rectangle of width w and length L

and a long, straight wire carrying a current I lie on a tabletop as

shown in Fig. 6. (a ) Determine the magnetic flux through the loop

due to the current I . (b ) Suppose the current is changing with time

according to I = a+ bt , where a and b are constants. Determine the

emf that is induced in the loop . What is the direction of the induced

current in the rectangle?

Problem 9 (10 points)

A rectangular loop of dimensions l and w moves with a constant

velocity v away from a long wire that carries a current I in the plane

of the loop (Fig. 7). 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.