A point charge q moves from point A to point D along the path ABCD (as shown) in a uniform electric field E pointing parallel to negative direction of X-axis. The co-ordinates of A, B, C, D are (0, 0, 0), (a, b, 0), (3a, 0, 0) and (2a, -b, 0) respectively. The work done by the field in the above process is given by expression:

$-2aqE$

The correct answer is Option (4) → $-2aqE$

To calculate the work done by a uniform electric field $(\vec E)$, when a point charge move along a given path,

$W=q\int\vec E.d\vec r$

$A(0,0,0)$

$B(a,b,0)$

$C(3a,0,0)$

$D(2a,-b,0)$

$W_{A→B}=q\vec E.d\vec r_{AB}$

$=q(-E\hat i).(a\hat i+b\hat j)$

$=-qEa$

$W_{B→C}=q(-E\hat i).(2a\hat i-b\hat j)$

$=-2qEa$

$W_{C→D}=q(-E\hat i).(-a\hat i-b\hat j)$

$=qEa$

$W=W_{AB}+W_{BC}+W_{CD}$

$=-2qEa$