Question

The work done to move a charge along an equipotential from A to B ______.

1. cannot be defined as `- int_A^B E.dl`
2. must be defined as `- int_A^B E.dl`
3. is zero.
4. can have a non-zero value.

Options

• a and b

• b and c

• c and d

• a and c

Answer 1

b and c

Explanation:

The work done by the external agent in shifting the test charge along the dashed line from 1 to 2 is

The external agent does a work `W = - q int_1^2 vecE * vec(dl)` in transporting the test charge q slowly from the positions 1 to 2 in the static electric field.

`W_("ext") = int_1^2 vecF_("ext") * vec(dl) int_1^2 (-qvecE) * vec(dl) = - q int_1^2 vecE * vec(dl)`

We know `V_A - V_B = - int_A^B vecE * vec(dl)`  ......(i)

`W_("electrical") = -ΔU = - qΔV = q(V_A - V_B)`  ......(ii)

Hence from (i) and (ii), `W_("electrical") = q(V_A - V_B) = - qint_A^B vecE * vec(dl)`

If we want to calculate the work done to move a charge along an equipotential from A to B

For equipotential surface V_A = V_B, hence W = 0

Also electric field is perpendicular to equipotential surface, hence `vecE * vec(dl) => W_("electrical") = 0`