Phys 742 – Assignment 2 – Due 6 Feb 96

1. (Similar to 3 of Qualifier file)

A circular ring of radius a carries a uniformly distributed charge q.

(a) Find the electrostatic potential on the axis of the ring.

(b) Find the potential at any point in space. You can express the result as an elliptic integral or a hypergeometric function (see notes), or as an expansion in Legendre polynomials (Jackson, page 91), but none of it is necessary to answer part (c).

(c) A test charge is located at the center of the ring. Is this a position of stable equilibrium? Consider both signs of the test charge. Find the restoring force for small displacements of the test charge.

(d) Suppose now that the ring is elliptical, or maybe a hexagon, or any shape with a center of symmetry. Is the center still a position of equilibrium and is the equilibrium stable, in general?

2. (Similar to Jackson’s 1.20)

(a) Give an approximate formula relating your electrical capacitance C, in centimeters, to your mass m, in kilograms. The formula should contain only known numbers, including the ratio of height h to girth G, or rather α = 2πh/G, which does not vary widely for most people of your age (what is it, numerically?). Give the numerical value of C for your present m.

(b) Similarly, relate numerically your C, in farads, to your weight w, in pounds. (To protect privacy, fictitious weight data will be accepted).

3. (Related to Jackson’s 1.8 and 1.9)

Consider a capacitor consisting of two concentric cylinders of length L, large compared to their radii a a, b (b > a). The potential difference between the cylinders is V.

(a) Compute Q, the magnitude of the charge per unit length on each cylinder.

(b) Compute the force acting on the unit area of each cylinder by using Jackson’s eq. (1.59)

(c) Compute (1/2πa)(∂W/∂a), where W is the energy per unit length in the capacitor. Is this the same as the force per unit area of the inner cylinder computed in (b)? Which, if any, of the four quantities (∂W/∂a)_{V} , −(∂W/∂a)_{V} , (∂W/∂a)_{Q}, − (∂W/∂a)_{Q} yield the correct force, in magnitude and sign? Same questions for the outer cylinder.

(d) Is the configuration stable with respect to off-center diplacement of the inner cylinder? Find the force as a function of displacement.

4. (Problem 1 of qualifier file)

A permanent electric dipole of moment p is situated at a distance r from a grounded conducting plane and its orientation forms an angle θ with the normal to the plane.

(a) What is the torque acting on the dipole?

(b) What is the stable orientation of the dipole?

(c) What is the force acting on the dipole in the stable equilibrium orientation?

Answer 3

(a) Q = CV with 1/C = 2 ln(b/a).

(b) For the inner cylinder, σ = Q/2πa and f = 2πσ^{2} = Q^{2}/2πa^{2}. For the outer cylinder, f = −Q^{2}/2πb^{2}.

(c) W = Q^{2}/2C = Q^{2} ln(b/a) and for the inner cylinder

f = −\frac{1}{2πa} (\frac{∂W}{∂a} )_{Q} = \frac{Q^{2}}{2πa^{2}}

(∂W/∂a)_{V} also gives the correct answer. Similarly for the outer cylinder

f = −\frac{1}{2πb} (\frac{∂W}{∂b} )_{Q} = −\frac{Q^{2}}{2πb^{2}}

(d)