Question

A clock face has negative charges $-q,-2q,-3q.....,-12q$ fixed at the position of the corresponding numerals on the dial. The clock hands do not disturb the net field due to point charges. At what time does the hour hand point in the same direction as electric field at the centre of the dial.

Answer 1

If the radius of clock will be $r$ then the electric field due to charges would be given by-

$\overrightarrow{E}=kQ/r^2(\cos \theta \hat{i}+\sin \theta \hat{j})$

Now for $i$th charge pointing at numeral $i$, we have

$Q=-iq$ and

$\theta =i\pi /6$

thus we have electric field due to $i$th charge as-

$\overrightarrow{E_i}=-kiq/r^2(\cos i\pi /6\hat{i}+\sin i\pi /6\hat{j})$

Now the net electric field at center would be given by

$\overrightarrow{E_{net}}=-kq/r^2{\sum }_1^{12}i(\cos i\pi /6\hat{i}+\sin i\pi /6\hat{j})$