On a clock face, point charges -q, -2q, -3q,..., -12q are fixed at the positions of the numerals 1, 2, 3,..., 12 respectively, all at the same distance from the centre. The clock hands themselves do not perturb the field. At what time does the hour hand point in the same direction as the n

Question

On a clock face, point charges -q, -2q, -3q,..., -12q are fixed at the positions of the numerals 1, 2, 3,..., 12 respectively, all at the same distance from the centre. The clock hands themselves do not perturb the field. At what time does the hour hand point in the same direction as the net electric field at the centre of the dial?

Accepted Answer

9:30. A negative charge produces a field at the centre that points toward the charge (toward its numeral). The magnitude due to numeral k is proportional to k. Taking the 12 (top) as +y and the 3 (right) as +x, numeral k sits in direction (sin(30k deg), cos(30k deg)). The resultant of Sum k*(sin30k, cos30k) over k=1..12 is a vector pointing at clock position 9.5, i.e. midway between the 9 and the 10. The hour hand points there at 9:30. (The equal background of -q at every numeral cancels by symmetry; only the linearly increasing 'extra' charge contributes, and its resultant points toward the 9-10 region.)

Solution

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