Question

A non-conducting sphere of radius $0.5\text{m}$ carries a total charge of $10\times {10}^{-10}\text{C}$ distributed uniformly which produces an electric field. Then the value of integral
$-\underset{\infty }{\overset{0}{\int }}\overrightarrow{E}\cdot \overrightarrow{dr}$ will be

• A. $27$
• B. $-27$
• C. $0$
• D. $-9$

Answer 1

The correct option is A $27$

Given that,
Radius of the sphere, $R=0.5\text{m}$
Charge of the sphere, $Q=10\times {10}^{-10}\text{C}$

The given integral represent the relation between electric field and electric potential.
$-\underset{\infty }{\overset{0}{\int }}\overrightarrow{E}\cdot \overrightarrow{dr}$ = Potential at centre of sphere - Potential at infinity

The potential at infinity is zero
$\Rightarrow V_{\infty }=0$
The potential at the centre of the sphere is
$V_{centre}=\frac{3kQ}{2R}$
$V_{centre}=\frac{3\times 9\times {10}^9\times 10\times {10}^{-10}}{2\times 0.5}=27$

So $-\underset{\infty }{\overset{0}{\int }}\overrightarrow{E}\cdot \overrightarrow{dr}=27$