Question

2.1 In this problem we find the electric field on the axis of a cylindrical shell of radius R and height h when the cylinder is uniformly charged with surface charge density . The axis of the cylinder is set on the z-axis and the bottom of the cylinder is set z = 0 and top z = h. We designate the point P where we measure the electric field to be z = z0. (See figure.) You will use the following formula for the magnitude of the electric field due to a ring (radius a) of uniform charge Q along the axis of the ring at distance x away: E = 1 4⇡✏0 Qx (x2 + a2) 3 2 (1) (The derivation of the above formula is found in Giancoli, Example 21-9.)

2.1 In this problem we find the electric field on the axis of a cylindrical shell of radius R and height h when the cylinder is uniformly charged with surface charge density σ. The axis of the cylinder is set on the z-axis and the bottom of the cylinder is set z 0 and top zh. We designate the point P where we measure the electric field to be z. (See figure.) You will use the following formula for the magnitude of the electric field due to a ring (radius a) of uniform charge Q along the axis of the ring at distance r away: Ta) 4me (The derivation of the above formula is found in Giancoli, Example 21-9.)

1. Before we begin the calculation, expect what the answer should look like:

(a) Which way is the direction of the electric field? Note, your answer may depend on the value of z0.

(b) Is there any point on the axis where you should expect the electric field to be zero? If so where? If not, why not?

(c) What should be the electric field approximately, when z0 >> h. Express the answer in terms of z0, R, h, and the fundamental constant epsilon0. (The solution should not be more than a couple of lines.)

2. What is the charge on a ring of width dz shown in the figure?

3. What is the distance to P from the center of the ring at z?

4. Find the electric field dE at P due to the ring. You may use the formula above but make sure to use the correct variables used in this problem.

5. Write out an integral expression for the total field at P, by summing the contributions from all the rings at di↵erent positions z that make up the entire cylindrical shell. Ensure that the limits on the integral are correct.

6. Evaluate the integral above to determine the total field at P, in terms of R, h, z0, and the fundametntal constant epsilon0. (Check that your expectations 1-(a) and 1-(b) are met.)

7. Expanding the result of 6 in power series of z0 and retaining the leading order, show that your result meets the expectation 1-(c) also.

Answer 1

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