The figures below depict two situations in which two infinitely l… - Vidyadip

MCQ

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\lambda$ are kept parallel to each other. In their resulting electric field, point charges $q$ and $- q$ are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement$(s)$ is(are)

A
Both charges execute simple harmonic motion.
B
Both charges will continue moving in the direction of their displacement.
✓
Charge $+ q$ executes simple harmonic motion while charge $- q$ continues moving in the direction of its displacement.
D
Charge $- q$ executes simple harmonic motion while charge $+ q$ continues moving in the direction of its displacement.

✓

Answer

Correct option: C.

Charge $+ q$ executes simple harmonic motion while charge $- q$ continues moving in the direction of its displacement.

c
In Case $I$ :

$\overrightarrow{ F }=\frac{\lambda q }{2 \pi \varepsilon_0( r + x )} \hat{ i }+\frac{\lambda q }{2 \pi \varepsilon_0( r - x )}(-\hat{ i })$

$=\frac{\lambda q }{\pi \varepsilon_0 I ^2} x (-\hat{ i })$

Hence $+ q$, charge will performs $S H M$ with time period $T =2 \pi \sqrt{\frac{ \pi r ^2 \varepsilon_0 m }{\lambda q }}$

In case II: Resultant force will act along the direction of displacement.

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