Question

Two equal charges q are kept fixed at a and +a along the $x-$ axis. A particle of mass m and change $\frac{q}{2}$ is brought to the origin and given a small displacement along the $x-$ axis, then:

• A. the particle executes oscillatory motion
• B. the particle remains stationary
• C. the particle executes, SUM along $x-$ axis
• D. the particle executes SHM along y-axis
• E. the particle moves on circular path

Answer 1

Answer: (C)

From Coulombs law
$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$ $F=\frac{1}{4\pi {\epsilon }_0}\frac{q\times q/2}{{(a+x)}^2}-\frac{1}{4\pi {\epsilon }_0}.\frac{q\times q/2}{{(a-x)}^2}$ $=\frac{1}{4\pi {\epsilon }_0}\frac{q^2}{2}\left[\frac{1}{{(a+x)}^2}-\frac{1}{{(a-x)}^2}\right]$ $=\frac{1}{4\pi {\epsilon }_0}\frac{q^2}{2}\left[\frac{4ax}{{(a^2-x^2)}^2}\right]$ When $x<<a,$ then $F=-\frac{2q^2}{4\pi {\epsilon }_0{a}^3}x$ $\Rightarrow$ $F\propto -x$ Hence, SHM along x-axis.