Question

Choose the correct:

Charge + q and - q are placed at points A and B respectively which are distance 2L apart. C is the midpoint of A and B. The work done in moving a charge + Q from C to D, along the semicircle CD as shown in the figure below, is ______.

विकल्प

• `(-qQ)/(6piε_0 L)`

• `(qQ)/(2piε_0 L)`

• `(qQ)/(6piε_0 L)`

• `(-qQ)/(4piε_0 L)`

Answer 1

Charge + q and - q are placed at points A and B respectively which are distance 2L apart. C is the midpoint of A and B. The work done in moving a charge + Q from C to D, along the semicircle CD as shown in the figure below, is `(-qQ)/(6piε_0 L)`.

Explanation:

Since the system consists of only conservative electrostatic forces, the work done in moving the +Q charge from point C to D is equal to the change in its electrostatic potential energy and is independent of path.

In the mathematical form, the statement can be written as follows:

W = U_D – U_C  .....(1)

Here, W is the work done in moving charge +Q from point C to D, U_D s the electrostatic potential energy at point D, and U_c is the electrostatic potential energy at point C.

The potential energy U of two point charges q₁ and q₂ can be written as: 

U = `(q_1q_2)/(4piε_0r)`

Here, ε₀ is the permittivity of vacuum, and r is the distance between the charges.

Value of U_D can be found as follows:

U_D = `(+ qQ)/(4piε_0 * (3L)) + (- qQ)/(4piε_0 * (L))`

U_D = `- (qQ)/(6piε_0 L)`  .....(2)

Value of U_C can be found as follows:

U_C = `(+ qQ)/(4piε_0 L) + (-qQ)/(4piε_0 L)`

U_C = 0  ......(3)

Substitute U_D from equation (2) and U_C from equation (3), in equation (1) to find W.

W = `- (qQ)/(6piε_0 L) - 0`

W = `- (qQ)/(6piε_0 L)`