Figure shown three points. `X, Y` and `Z` forming an equilaternal triangle of side `s` in an unifrom electric field of strength `E`. A unit positive test charge is moved from `X` to `Y`, from `Y` to `Z`, and from `Z` back to `X`.

Which one of the following correctly gives the work done against electrical forces in moving the charge along the various parts o fthis path ?

Options

A. `{:("X to Y"," Y to Z"," Z to X",),(+Es,+"Es "cos 60^(@).,-"Es "cos 60^(@).,):}`
B. `{:("X to Y"," Y to Z"," Z to X",),(0,-"Es "cos 60^(@).,+"Es "cos 60^(@).,):}`
C. `{:("X to Y"," Y to Z"," Z to X",),(+Es,+"Es "cos 60^(@).,-"Es "cos 60^(@).,):}`
D. `{:("X to Y"," Y to Z"," Z to X",),(0,-"Es "sin 60^(@).,+"Es "sin 60^(@).,):}`

Answer

D

Explanation

From `E = (dV)/(dx)`, work done in moving a unit positive test charge in an unifrom electric field of strength `E` is given by
work done `V = int dV = - int Edx`
`= - E int dx = - Ed`
where `d` is the distancetravelled parallel to the direction of `E`.
From `X` to `Y, d = 0`. Therefore work done `= 0`
From `Y` to `Z, d = +s sin 60^(@)`
Therefore work done is `Es sin 60^(@)`
From `Z` to `X, d = - x sin 60^(@)`
Therefore work done is `- Es sin 60^(@)`