A long, thin, nonconducting plastic rod is bent into a loop with radius R. Betwe

Question

A long, thin, nonconducting plastic rod is bent into a loop with radius R. Between the ends of the rod, a small gap of length ( << R) remains. A charge Q is equally distributed on the rod.
(a) Indicate the direction of the electric field at the center of the loop. (Select all that apply.)
If Q is positive, the field at the origin points radially outward.
The field at the origin always points radially inward.
If Q is positive, the field at the origin points radially inward.
If Q is negative, the field at the origin points radially inward.
If Q is negative, the field at the origin points radially outward.
The field at the origin always points radially outward.


(b) Find the magnitude of the electric field at the center of the loop. (Use any variable or symbol stated above along with the following as necessary: k.)
Ecenter =

Please explain in detail, thanks in advance

Explanation / Answer

Given the total charge of the rod = Q. Let the total length of the rod be 'L' and let 'x' be the length of the short gap between the two ends of the loop. a) Since we are considering the field direction at the centre of the loop, let us divide the loop into infinitisimal parts each of length dx. By symmetry, we can observe that for any given part dx, there will be radially opposite part dx' of equal size (and hence equal charge) and at same distance from centre but obviously exerting an electric field in opposite direction. Hence these two fields cancel each other and so they doesn't contribute to any resultant field. However, there is a part of length 'x' left over in the loop exactly on the opposite side of the gap which cannot be cancelled (since gap do not have any charge). Hence the resultant field will be due to this small part left over and obviously away from this part, i.e., towards the gap. b) Now the magnitude of this electric field is given by       E = 9 * 10^9 * q / a^2       (net charge of this part with length x = charge density * length = Q/L * x)        = 9 * 10^9 * (Q/L) * x / a^2   N/C        (I hope you can draw the diagram and follow my argument to understand the problem better. There is some technical problem due to which I am unable to add pictures). Hope this helps you.

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