Question

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 continue to move in the direction of their displacement
C) charge +q executes simple harmonic motion while -q continues moving in direction of its displacement
D) none

Answer 1

let us find the net force acting on the point charges due to the two infinitely long static line charges. In the first case, we need to assume the charge q as positive and find the relation between the force acting on the charge and the distance x. Next, we need to find the same for charge -q.

Complete step-by-step answer:
let us take the first case as charge q is positive,
The linear charge density is given as $\lambda$, the movement of the two given charges is only in the direction of x.
Now,
Q is positive:

Let the distance between the line charges be d and the charge q be displaced by a small distance x.
Q will be repelled more by the right line charge than the left line charge. The net force will be towards left.

$\begin{aligned} & {{F}_{q}}=\dfrac{2kq\lambda }{d+x}\hat{i}+\dfrac{2kq\lambda }{d-x}(-\hat{i}) \\\ & \Rightarrow {{F}_{q}}=\dfrac{4kqx\lambda }{{{d}^{2}}-{{x}^{2}}}(-\hat{i}) \\\ & \Rightarrow {{F}_{q}}=\dfrac{4kq\lambda }{{{d}^{2}}}(-\hat{i}),x < < d \\\ \end{aligned}$
Thus, force on q is directly proportional to displacement x.

When charge q is negative:
When displaced by x towards right, the force of attraction due to right line charge will be more than the force of attraction due to left line charge and the charge will accelerate towards right.
If we consider the situations, it is clear that the charge +q executes simple harmonic motion while the charge -q, on the other hand, continues to move in the direction of displacement.
Therefore, the correct option is option c.

So, the correct answer is “Option C”.

Additional Information: The repulsive or attractive interaction between any 2 charged bodies is called an electric force. Similar to any force, its impact and effects on the given body are described by Newton's laws of motion. Be electrical the electric force item on the list of other forces that exist over objects. Newton’s laws are applicable to analyse the motion under the influence of that kind of force or combination of forces. He analysis begins by the construction of a free body image wherein the direction and type of the individual forces are shown by the vector to calculate the resultant sum which is called the net force that can be applied to determine the body’s acceleration.

Note: The electric field gets affected due to the presence of the charges that are outside the closed surface too. Therefore, we need to consider the electric charges present outside the closed surface and also inside the closed surface.