Question

Two spherical conductors a1 and a2 of radii r1 and r2 are placed concentrically in air. The two are connected by copper wire as shown in figure. Then equivalent capacitance of the system is

Answer 1

Given:

Two spherical conductors of radii r1 and r2 are placed concentrically in air. The two spheres are connected by a copper wire.

To find:

Equivalent capacitance of the system.

Concept:

Charges always have a tendency to be spread over a larger area. Hence whenever the smaller sphere is connected to the larger concentric sphere by a conductor , all the charges on the smaller sphere go to the larger sphere.

Hence , net capacitance will be equal to the capacitance of the larger sphere.

Calculation:

Let smaller sphere be a1 , having charge q1 and radius r1 ; similarly larger sphere be a2 , having charge q2 and radius r2.

(Refer to the diagram)

After connecting with copper wire , charge on sphere a2 will be $q1 + q2$.

Potential on surface of larger sphere be V

$\therefore \: \sf{V = \dfrac{k(q1 + q2)}{r2} }$

Let net capacitance be C

$\therefore \sf {C = \dfrac{Q}{V} }$

$= > \sf {C = \dfrac{(q1 + q2)}{ \bigg \{ \frac{k(q1 + q2)}{r2} \bigg\} } }$

$= > \sf {C = \dfrac{r2}{k}}$

Putting value of Coulomb's Constant :

$= > \sf {C = \dfrac{r2}{ \{\frac{1}{ 4\pi\epsilon_{0}} \}}}$

$= > \sf {C = 4\pi\epsilon_{0}(r2)}$

So final answer is :

$\boxed{\bold {C = 4\pi\epsilon_{0}(r2)}}$