Physics 222 Quiz 4 Spring 2014

Quiz 4A A solid conducting sphere with radius 1 cm, that carries a charge of +5 nC, is concentric with a very thin insulating shell of radius 2 cm that also carries a charge +5 nC, which is distributed uniformly over the insulating shell. Find the electric field magnitude in each of the following regions:

a.) 0 < r < 1 cm; E = 0 in this region since r is inside the conducting sphere.

b.) 1 cm < r < 2 cm; The charge enclosed is +5 nC; by Gauss’s Law, this will be equivalent to having a +5 nC point charge at the origin. We will thus find that E = kq/r2 = (8.99 * 109 N m2/C2) * (5 * 10-9 C) / r2 = 44.95/r2 N/C

c.) r > 2 cm. Now the charge enclosed is double that of part (b), and the result is equivalent to that of a 10 nC point charge at the origin. The result is thus twice that of part (b): E = 89.9/r2 N/C.

Physics 222 Quiz 4 Spring 2014

Quiz 4B A solid conducting cylinder with radius 1 cm (and infinite length), that carries a charge per unit length of +5 nC/m, is coaxial with a very thin insulating cylindrical shell of radius 2 cm that also carries a charge per unit length +5 nC/m, which is distributed uniformly over the insulating shell. Find the electric field magnitude in each of the following regions, where r is the distance away from the central axis:

a.) 0 < r < 1 cm; E = 0 in this region since r is inside the conducting cylinder.

b.) 1 cm < r < 2 cm; For a 1 m long Gaussian cylinder, the charge enclosed is +5 nC; by Gauss’s Law, this will be equivalent to having a +5 nC/m line charge on the central axis. We will thus find (using the line charge formula) that: E = k * 2λ/r = (8.99 * 109 N m2/C2) * 2 * (5 * 10-9 C/m) / r = 89.9/r N/C

c.) r > 2 cm. Now the charge enclosed is double that of part (b), and the result is equivalent to that of a 10 nC/m line charge on the central axis. This is twice the charge density of part (b), so we expect that E = 179.8/r N/C.

Physics 222 Quiz 4 Spring 2014

Quiz 4C

A solid conducting sphere with radius 2 cm, that carries a charge of +3 nC, is concentric with a very thin insulating shell of radius 4 cm that carries a charge −3 nC, which is distributed uniformly over the insulating shell. Find the electric field magnitude in each of the following regions:

a.) 0 < r < 2 cm; E = 0 in this region since r is inside the conducting sphere.

b.) 2 cm < r < 4 cm; The charge enclosed is +3 nC; by Gauss’s Law, this will be equivalent to having a +3 nC point charge at the origin. We will thus find that E = kq/r2 = (8.99 * 109 N m2/C2) * (3 * 10-9 C) / r2 = 26.97/r2 N/C

c.) r > 4 cm. Here the total charge enclosed is again zero, so once more, E = 0.

Physics 222 Quiz 4 Spring 2014

Quiz 4D

A solid conducting cylinder with radius 2 cm (and infinite length), that carries a charge per unit length of +3 nC/m, is coaxial with a very thin insulating cylindrical shell of radius 4 cm that carries a charge per unit length −3 nC/m, which is distributed uniformly over the insulating shell. Find the electric field magnitude in each of the following regions, where r is the distance away from the central axis:

a.) r < 2 cm; E = 0 in this region since r is inside the conducting cylinder.

b.) 2 cm < r < 4 cm; For a 1 m long Gaussian cylinder, the charge enclosed is +3 nC; by Gauss’s Law, this will be equivalent to having a +3 nC/m line charge on the central axis. We will thus find (using the line charge formula) that: E = k * 2λ/r = (8.99 * 109 N m2/C2) * 2 * (3 * 10-9 C/m) / r = 53.94/r N/C

c.) r > 4 cm. Here the total charge enclosed sums to zero, so again E = 0.