Charge – Q and 2Q are distributed uniformly on surface of two concentric spherical shells of radii ‘R’ and ‘2R’ respectively as shown in the figure. Select correct alternative(s) (A) the total electrostatic energy stored in the system is Q2 / 8πε0 R (B) electrostatic energy in the space between two shells is Q2 / 16πε0 R (C) electrostatic energy stored outside the system is Q2 / πε0 R (D) electrostatic energy in space between two shells is zer
Question
Charge – Q and 2Q are distributed uniformly on surface of two concentric spherical shells of radii ‘R’ and ‘2R’ respectively as shown in the figure. Select correct alternative(s) (A) the total electrostatic energy stored in the system is Q2 / 8πε0 R (B) electrostatic energy in the space between two shells is Q2 / 16πε0 R (C) electrostatic energy stored outside the system is Q2 / πε0 R (D) electrostatic energy in space between two shells is zer
Solution
The electrostatic energy stored in a system of charges is given by the formula:
U = 1/2 ∫ ρV dτ
where ρ is the charge density, V is the potential due to all other charges, and the integral is over all space.
(A) The total electrostatic energy stored in the system is Q^2 / 8πε0 R
The total energy stored in the system is the sum of the energy stored in each shell. The energy stored in a shell of charge Q and radius R is given by U = Q^2 / 8πε0 R. The energy stored in a shell of charge 2Q and radius 2R is given by U = 4Q^2 / 8πε0 R = Q^2 / 2πε0 R. Therefore, the total energy stored in the system is Q^2 / 8πε0 R + Q^2 / 2πε0 R = 5Q^2 / 8πε0 R. So, option (A) is incorrect.
(B) The electrostatic energy in the space between two shells is Q^2 / 16πε0 R
The energy in the space between the shells is the difference of the total energy and the energy of the outer shell. So, it is 5Q^2 / 8πε0 R - Q^2 / 2πε0 R = Q^2 / 8πε0 R. So, option (B) is incorrect.
(C) The electrostatic energy stored outside the system is Q^2 / πε0 R
The energy stored outside the system is the energy of the outer shell, which is Q^2 / 2πε0 R. So, option (C) is incorrect.
(D) The electrostatic energy in space between two shells is zero
The energy in the space between the shells is not zero, as calculated in option (B). So, option (D) is incorrect.
Therefore, none of the given options are correct.
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