Question

A rod of length $l$ located along the x-axis has a total charge $Q$. Find the electric potential at a point P located on the y-axis at a distance $a$ from the origin.

• A. $\frac{Q}{\pi {ϵ}_{0}l}\ln \frac{l+\sqrt{l^2-a^2}}{a}$
• B. $\frac{Q}{\pi {ϵ}_{0}l}\ln \frac{l-\sqrt{l^2+a^2}}{a}$
• C. $\frac{Q}{\pi {ϵ}_{0}l}\ln \frac{l-\sqrt{l^2-a^2}}{a}$
• D. $\frac{Q}{\pi {ϵ}_{0}l}\ln \frac{l+\sqrt{l^2+a^2}}{a}$

Answer 1

The correct option is D $\frac{Q}{\pi {ϵ}_{0}l}\ln \frac{l+\sqrt{l^2+a^2}}{a}$

${dV}_P=\frac{Kdq}{\sqrt{x^2+a^2}}$

${dV}_P=\frac{K\lambda dx}{\sqrt{x^2+a^2}}$

${dV}_P=\frac{KQ}{L}{\int }_0^l\frac{1}{\sqrt{x^2+a^2}}dx$

$V_P=\frac{KQ}{L}{\left[In\left(\frac{x+\sqrt{x^2+a^2}}{a}\right)\right]}_0^l$

$[V_P=\frac{KQ}{L}In\left(\frac{l+\sqrt{l^2+a^2}}{a}\right)]$

potential at point P

Option (D) is correct.