A charge Q is distributed over two concentric conducting thin spherical shells radii r and R ( R r ). If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

Question

A charge Q is distributed over two concentric conducting thin spherical shells radii r and R ( R > r ). If the surface charge densities on the two shells are equal, the electric potential at the common centre is : (A) (1/4 π ε0) (( R +2 r ) Q /2( R 2+ r 2)) (B) (1/4 π ε0) (( R + r )/2( R 2+ r 2)) Q (C) (1/4 π ε0) (( R + r )/( R 2+ r 2)) Q (D) (1/4 π ε0) ((2 R+r)/(R2+r2)) Q

Tardigrade Admin · Accepted Answer

Let the charges on inner and outer spheres are

Q_{1}

and

Q_{2}

Since charge density '

σ^{'}

is same for both spheres, so

σ = \frac{Q _{1}}{4 π r ^{2}} = \frac{Q _{2}}{4 π R ^{2}} \Rightarrow \frac{Q _{1}}{Q _{2}} = \frac{r ^{2}}{R ^{2}}

Q_{1} + Q_{2} = Q \Rightarrow \frac{Q _{2} r ^{2}}{R ^{2}} + Q_{2} = Q

\Rightarrow Q_{2} = \frac{Q R ^{2}}{( r ^{2} + R ^{2} )}

Q_{1} = \frac{r ^{2}}{R ^{2}} \cdot \frac{Q R ^{2}}{( R ^{2} + r ^{2} )} = \frac{Q r ^{2}}{( R ^{2} + r ^{2} )}

Potential at centre

^{'} O^{'} = \frac{k Q _{1}}{r} + \frac{k Q _{2}}{R}

= k [\frac{Q r ^{2}}{r ( R ^{2} + r ^{2} )} + \frac{Q R ^{2}}{R ( R ^{2} + r ^{2} )}]

= \frac{k Q ( r + R )}{( R ^{2} + r ^{2} )} = \frac{1}{4 π ϵ _{0}} \frac{( R + r )}{( R ^{2} + r ^{2} )} Q

Q. A charge $Q$ is distributed over two concentric conducting thin spherical shells radii $r$ and $R$ $(R > r)$ . If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

$\frac{1}{4 π ε _{0}} \frac{( R + 2 r ) Q}{2 ( R ^{2} + r ^{2} )}$

$\frac{1}{4 π ε _{0}} \frac{( R + r )}{2 ( R ^{2} + r ^{2} )} Q$

$\frac{1}{4 π ε _{0}} \frac{( R + r )}{( R ^{2} + r ^{2} )} Q$

$\frac{1}{4 π ε _{0}} \frac{( 2 R + r )}{( R ^{2} + r ^{2} )} Q$

Q. A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R>r). If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

4πε0​1​2(R2+r2)(R+2r)Q​

4πε0​1​2(R2+r2)(R+r)​Q

4πε0​1​(R2+r2)(R+r)​Q

4πε0​1​(R2+r2)(2R+r)​Q

Q. A charge $Q$ is distributed over two concentric conducting thin spherical shells radii $r$ and $R$ $(R > r)$ . If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

$\frac{1}{4 π ε _{0}} \frac{( R + 2 r ) Q}{2 ( R ^{2} + r ^{2} )}$

$\frac{1}{4 π ε _{0}} \frac{( R + r )}{2 ( R ^{2} + r ^{2} )} Q$

$\frac{1}{4 π ε _{0}} \frac{( R + r )}{( R ^{2} + r ^{2} )} Q$

$\frac{1}{4 π ε _{0}} \frac{( 2 R + r )}{( R ^{2} + r ^{2} )} Q$