1. Potential Due to the Shell
* Inside the shell (r < R): The electric field inside a uniformly charged spherical shell is zero. Therefore, the potential is constant and equal to the potential on the surface of the shell.
* Outside the shell (r > R): The electric field outside the shell is the same as that of a point charge Q located at the center of the shell. Using Coulomb's Law, the potential at a distance r from the center is:
V(r) = kQ/r
where k is Coulomb's constant (1/4πε₀).
2. Calculating the Energy
The energy stored in a charged system can be calculated using the following approach:
* Energy = Work done to assemble the charge
Imagine building up the charge on the shell gradually. At any moment, the potential due to the charge already on the shell is V(R) = kQ/R. To bring in an infinitesimal amount of charge dQ, the work done is:
dW = V(R) dQ = (kQ/R) dQ
To find the total energy, we integrate this expression from zero charge to the final charge Q:
U = ∫dW = ∫₀^Q (kQ/R) dQ = (k/R) ∫₀^Q Q dQ
U = (k/R) * (Q²/2)
Therefore, the energy of a uniformly charged spherical shell is:
U = (kQ²/2R) = (Q²/8πε₀R)
Key Points
* Symmetry: The spherical symmetry is crucial. The electric field and potential have simple expressions due to this symmetry.
* Method of Assembly: The energy calculation relies on the idea of gradually assembling the charge, which allows us to use the potential at each step to calculate the work done.
* Potential Energy: The energy stored in the charged shell represents the potential energy of the system due to the electrostatic forces between the charges.
