Non uniform linear charge density?

fardeen_gen · #1 $Old$ June 29th, 2009, 03:58 PM

The plastic rod of length $L$ in the diagram has non uniform linear charge density $\lambda = cx$ where c is a positive constant.

(a) With V = 0 at infinity, find the electric potential at point $P_2$ on the y-axis, at a distance y from one end of the rod.

(b) From that result, find the electric field component $E_y$ at $P_2$

(c) Why cannot the field component $E_x$ at $P_2$ be found using the result of (a)?

malaygoel · #2 $Old$ June 30th, 2009, 06:36 AM

Quote:

Originally Posted by fardeen_gen $View Post$

The plastic rod of length $L$ in the diagram has non uniform linear charge density $\lambda = cx$ where c is a positive constant.

(a) With V = 0 at infinity, find the electric potential at point $P_2$ on the y-axis, at a distance y from one end of the rod.

(b) From that result, find the electric field component $E_y$ at $P_2$

(c) Why cannot the field component $E_x$ at $P_2$ be found using the result of (a)?

Considering small length $dx$ of the rod, the charge $dq$ will be given by $\lambda dx$

Hence, potential at point $P_2$ will be given by
$\int_0^L \frac{dq}{4\pi\epsilon_0 r}$

where 'r' is the distance of the small element from $P_2$

$E_x$ could not be calculated from expression for potential derived in (a) because it requires the expression to be a function of x whereas in (a) it has been derived as function of y.