There is a uniform electrostatic field in a region. The potential at various points on a small sphere centred at $P$, in the region, is found to vary between the limits $589.0 \mathrm{V}$ to $589.8 \mathrm{V}$. What is the potential at a point on the sphere whose radius vector makes an angle of $60°$ with the direction of the field?

• A

  $589.4 \mathrm{V}$
• B

  $589.5 \mathrm{V}$
• C

  $589.2 \mathrm{V}$
• Correct answer
  D

  $589.6 \mathrm{V}$

Solution

$\mathrm{\Delta }V=Ed$

$\mathrm{Along\; diameter}\mathrm{ }∆\mathrm{V}=589.8-589.0\mathrm{ }\mathrm{ }\mathrm{V}$
                              $=0.8\mathrm{ }\mathrm{V}$

$0.8=Ed$

Now

$\Delta V=E\frac{d}{2}\cos \theta =0.4\times \cos {60}^o$

$=0.2$

$\therefore 589.8-V=0.2$

$V=589.6 \mathrm{V}$

Hence, the correct answer is option: 'D'.