Answer to Question #97430 in Electricity and Magnetism for shah

Question

(a) Twelve equal charges, q, are situated at the corners of a regular 12–sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center? (b) Suppose one of the 12 q’s is removed (the one at “6 o’clock”). What is the force on Q? Explain your reasoning carefully (c) Now 13 equalcharges, q,areplacedatthecornersofaregular 13–sidedpolygon. Whatisthe force on a test charge Q at the center? (d) If one of the 13 q’s is removed, what is the force on Q? Explain your reasoning.

Answer 1

I should correct the answer d):

d) if one charge is removed, an uncompensated tension arises, which is opposite in the direction to the field strength of the removed charge, that is, the resulting force is equal $f = \frac{kqQ}{R^2}$ , where R is the radius of the circumscribed circle. The result can be obtained by direct calculation: force $F = \frac{2kqQ}{R^2}(cos\alpha + cos2\alpha + ... + cos6\alpha) = \frac{kqQ}{R^2}$ , where the angle $\alpha = \frac{360}{13}.$