Module 2 : Electrostatics
Lecture 8 : Electrostatic Potential
 

Example 13 :

Obtain an expression for the potential at a distance $x$ from a line charge distribution with a linear charge density $\lambda$.

Solution

The electric field due to the charge distribution at a point P located at a distance $x$is

\begin{displaymath}\vec E = \frac{\lambda}{2\pi\epsilon_0}\frac{1}{x}\hat n\end{displaymath}

where $\hat n$ is along the perpendicular from the point P to the line charge, as shown (the direction is opposite if line charge density is negative). The potential difference between the point $P$and a reference point $P_0$is obtained by calculating the value of the integral $\int \vec E\cdot \vec{dl}$from the point $P_0$to the point $P$. As the integral is independent of path, we calculate it along the path $P_0\rightarrow P_1\rightarrow P$, as shown. The contribution to the integral from the path $P_0\rightarrow P_1$is zero as along this path $\vec{dl}$ is perpendicular to $\vec E$. Along the path $P_1P$, the directions of $\vec{dl}$ and $\vec E$ are parallel.

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