Question

What is the amount of work done in moving a point charge Q around a circular arc of radius ‘r’ at the center of which another point charge ‘q’ is located?

Answer 1

When the charge Q moves in the circular arc around charge q, distance of Q from q remains constant so potential energy of the two pair charge system remains constant. Again, we know that the work done is given as the scalar product of the force and the displacement.

Formula used:
The work done is given by the formula:
$W = Fd\cos \theta$
Where, $F$ is the centripetal force, $d$ is the distance between the two charges, $\theta$ is the angle between the force and the displacement.

Complete step by step solution:
Let’s define the circle first to get more clarity. A circle is a shape consisting of all points in a plane that are a given distance from a given point, the center; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
As we can see from the above definition, the distance of the moving point from the center remains constant.

So, in this case distance of charge Q remains constant with respect to the fixed charge q as long as Q rotates in the circular arc and distance is r. Now potential energy only depends upon distance between the charges only.Since distance is not changing so the potential energy of the two-particle system given in the question will not change. Or there is no change in potential energy.

Since there is no change in potential energy there will be no work done. We can also find out the amount of work by using the formula: When the charge is making circular motions, then the displacement and the centripetal force are aligned perpendicular to each other.

W = Fd\cos \theta \\\ \Rightarrow W = Fd\cos 90^\circ \\\ \Rightarrow W = Fd \times 0 \\\ \Rightarrow W = 0 \\\ $$ **Hence the correct answer is zero joules.** **Note:** While solving this problem, most of the students can interpret the angle between the centripetal force and the displacement. It is important to remember that the centripetal force experienced by the charge which is moving in circular motion, is directed inwards. The force and the direction displacement are perpendicular to each other. The direction of displacement in circular motion is along the tangent drawn at that point.