Question

Two isolated metallic solid spheres of radii R and 2R are charged such that both of these have same charge density $\sigma$.The spheres are located far away from each other and connected by a thin conducting wire. Find the new charge density on the bigger sphere.

Answer 1

Answer:

The new charge density on the bigger sphere is 5/6 ρ

Given:

The radius of 1st metallic solid sphere is R.

The radius of 2nd metallic solid sphere is 2R.

Let the potential difference of the charges be V.

Let the initial charge on the smaller sphere be q1= $4 \pi R^{2} \rho$

Let the initial charge on the longer sphere be Q1=$16 \pi R^{2} \rho$

We know that,

The capacitance of the sphere α to its radius.

Therefore the capacitance will be in the ratio of R:2R = 1:2

Let the capacitance of the smaller be Cs =C

Let the capacitance of the longer be Cl =2C

We know that,CsV+ClV=q1+Q1= $20 \pi R^{2} \rho$

By substituting the values for Cs and Cl we get,

$=\mathrm{CV}+2 \mathrm{CV}=20 \pi R^{2} \rho$

To find the potential difference,

$\mathrm{V}=\frac{20 \pi R^{2} \rho}{3 C}$

Therefore the longer capacitor as the changed charge as,

$\begin{array}{l}{C_{l} V=2 \mathrm{C} \times \frac{20 \pi R^{2} \rho}{3 c}} \\ {=\frac{40 \pi R^{2} \rho}{3 c}}\end{array}$

Hence the charge density= $\frac{20 \pi R^{2} \rho}{3 \times 16 \pi R^{2} \rho}$

=5/6 ρ

Answer 2

The new charge density on the bigger sphere is 5/6 ρ

Given:

The radius of 1st metallic solid sphere is R.

The radius of 2nd metallic solid sphere is 2R.

Let the potential difference of the charges be V.

Let the initial charge on the smaller sphere be $q1= 4 \pi R^{2}$

Let the initial charge on the longer sphere be $Q1=16 \pi R^{2}$

We know that,

The capacitance of the sphere α to its radius.

• Therefore the capacitance will be in the ratio of R:2R = 1:2

• Let the capacitance of the smaller be Cs =C

• Let the capacitance of the longer be Cl =2C

We know that,$CsV+ClV=q1+Q1= 20 \pi R^{2}$

By substituting the values for Cs and Cl we get,

$=\mathrm{CV}+2 \mathrm{CV}=20 \pi R^{2}$

• Hence the charge density==5/6 ρ