A thin glass rod of length has a linear charge density that is zero in the middl

Question

A thin glass rod of length has a linear charge density that is zero in the middle of the length of the rod and increases linearly (both positively) along the length of the rod in both directions from the middle of the rod.

a) What is the electric field at a distance y above the middle of the rod?

b) If the rod extends infinitely in both directions, find the electric field at a distance y above the middle of the rod. (directly above where the charge density is zero).

Please DO NOT use a calculator and show all work!! Thank you

Explanation / Answer

Here lambda = ax

E = 2* integral k dq/(x^2 +y^2) cos theta

= 2k * integral ax dx/(x^2 +y^2) y/sqrt(x^2 +y^2)

Let (x^2 +y^2) = r

2x dx = dr

E = ak y* integral dr/r^1.5

= aky r^-0.5/-0.5

= - 2 aky/sqrt(x^2 +y ^2) from x=0 to x=0.5L

= 2aky *(1/y - 1/sqrt(y^2 + 0.25 *L^2)) where k is coloumbs constant.

B) if L is infinity,

E = 2a k where k is coloumbs constant

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