By Hermann A. Haus
Haus H.A., Melcher J.R. Electromagnetic fields and effort (PH, 1989)(ISBN 013249020X)(863s)_PE_
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Additional info for Electromagnetic fields and energy
7. For an observer at the radius r, translation of the line source in the z direction and rotation of the source about the z axis (in the φ direction) results in the same charge distribution, so the electric field must only depend on r. Moreover, E can only have a radial component. To see this, suppose that there were a z component of E. Then a 180 degree rotation of the system about an axis perpendicular to and passing through the z axis must reverse this field. However, the rotation leaves the charge distribution unchanged.
The objective is to measure this charge, q, without removing it from the ball. We know from the discussion of Gauss’ law in Sec. 3 that this charge is the source of an electric field. In general, this field terminates on charges of opposite sign. Thus, the net charge that terminates the field originating from q is equal in magnitude and opposite in sign to q. Measurement of this “image” charge is tantamount to measuring q. How can we design a metal electrode so that we are guaranteed that all of the lines of E originating from q will be terminated on its surface?
6 is found to be given by (12). Evaluation of the left-hand side of (1) over any other surface that encloses the point charge must also give q. Suppose that the closed surface S is composed of a hemisphere of radius a in the upper half-plane, a hemisphere of radius b in the lower half-plane, and a washer-shaped flat surface that joins the two. In spherical coordinates (defined in Table I), these three parts of the closed surface S are defined by (r = a, 0 < θ < 12 π, 0 ≤ φ < 2π), (r = b, 21 π < θ < π, 0 ≤ φ < 2π), and (θ = 12 π, b ≤ r ≤ a, 0 ≤ φ < 2π).
Electromagnetic fields and energy by Hermann A. Haus