Question

Charge Q coulombs is uniformly distributed throughout the volume of a solid hemisphere of radius R m. Then the potential ar centre O of the hemisphere in volts is

Answer 1

Given:

Charge Q coulombs is uniformly distributed throughout the volume of a solid hemisphere of radius R .

To find:

Potential at centre of hemisphere

Calculation:

The general formula for Electrostatic Potential in a solid sphere at a distance r < R is :

$\boxed{ \sf{V = \dfrac{kQ}{2R} \bigg \{3 - \dfrac{ {r}^{2} }{ {R}^{2} } \bigg \}}}$

At the centre of the sphere , r = 0 ;

$\sf{ \therefore \: V = \dfrac{kQ}{2R} \bigg \{3 - \dfrac{ {r}^{2} }{ {R}^{2} } \bigg \}}$

$\sf{ = > \: V = \dfrac{kQ}{2R} \bigg \{3 - \dfrac{ {(0)}^{2} }{ {R}^{2} } \bigg \}}$

$\sf{ = > \: V = \dfrac{kQ}{2R} \bigg \{3 - 0\bigg \}}$

$\sf{ = > \: V = \dfrac{3kQ}{2R} }$

Putting value of Coulomb's Constant :

$\sf{ = > \: V = \dfrac{3Q}{4\pi\epsilon_{0} (2R)} }$

$\sf{ = > \: V = \dfrac{3Q}{8\pi\epsilon_{0} R} }$

So, final answer is:

$\boxed{ \red{ \large{\rm{ \: V = \dfrac{3Q}{8\pi\epsilon_{0} R} }}}}$