Physics Problems & Solutions: Electromagnetism

Two positive charged of q and 2q coulombs are located on the x-axis at x = 0.5a and 1.5a respectively, as shown above. There is an infinite grounded conducting plane at x = 0. What is the magnitude of the net force on the charge q?

A. $\frac{1}{4\pi \epsilon_0} \frac{q^2}{a^2}$

B. $\frac{1}{4\pi \epsilon_0} \frac{3q^2}{2a^2}$

C. $\frac{1}{4\pi \epsilon_0} \frac{2q^2}{a^2}$

D. $\frac{1}{4\pi \epsilon_0} \frac{3q^2}{a^2}$

E. $\frac{1}{4\pi \epsilon_0} \frac{7q^2}{2a^2}$

(GR9277 #10)

Solution:

Using method of image:

On q:

$\begin{aligned} \vec{F}_{\text{total}}&=kq\left [ \frac{2q}{a^2}\left ( -\hat{x} \right )+ \frac{-q}{a^2}\hat{x} +\frac{-2q}{(2a)^2}\hat{x} \right ] \\&=\frac{kq^2}{a^2}\left ( -2-1-\frac{1}{2} \right )\hat{x} \\&=-\frac{7}{2}\frac{kq^2}{a^2}\hat{x} \\\left |\vec{F}_{\text{total}} \right |&=k\frac{7q^2}{2a^2}\end{aligned}$

with $k=\frac{1}{4\pi \epsilon_0}$

Answer: E

Physics Problems & Solutions

Electromagnetism - Method of Image

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