Question

Calculate the x-component of the electric field produced by the charge distribution Q at points on the positive x-axis where x>a. Express your answer in terms some or all of the variables Q, a, x, r, and constant k.

Answer 1

Answer:

Explanation:

From the image attached below:

It is required to determine the electric field because of the line charge distribution of positive charge distributed uniformly from x = 0 to x = a at point x from O;

where x > a

Assume we chose an element dy at a distance from the point.

Then, the change on $dy = \dfrac{Q}{a} \times dy$

The electric field due to this dy length is $\dfrac{kdq}{y^2 }= \dfrac{kQ dy }{ay^2}$

Thus, the total electric field $= \dfrac{kQ}{a}\int \limits ^{x}_{x-a} \dfrac{dy}{y^2}$

$= \dfrac{kQ}{a} \Big [ \dfrac{1}{y}\Big ]^{x-a}_{x}$

$=\dfrac{kQ}{a}\Big [\dfrac{1}{x-a}-\dfrac{1}{x}\Big ]$

$E=\dfrac{kQ}{a}\Big (\dfrac{x-(x-a)}{(x-a)x}\Big )$

$E=\dfrac{kQa}{a(x-a)x}$

Hence, the electric field  $E=\dfrac{kQ}{(x-a)x} \ \ \ where; x>a$