A spherical conducting shell has an inner radius \(r_1\) and an outer radius \(r_2,\) carrying a total charge \(Q.\) If a point charge \(q\) is placed at the center of this shell, what are the surface charge densities on the inner and outer surfaces of the shell, and is the electric field inside a charge-free cavity within any conductor in electrostatic equilibrium always zero, regardless of the conductor's shape?
| 1. | \(\sigma_{in}=\dfrac{-q}{4\pi r_1^2},~\sigma_{out}=\dfrac{Q+q}{4\pi r_2^2},\) (Yes, due to electrostatic equilibrium) |
| 2. | \(\sigma_{in}=\dfrac{q}{4\pi r_1^2},~\sigma_{out}=\dfrac{Q-q}{4\pi r_2^2},\) (No, only for spherical conductors) |
| 3. | \(\sigma_{in}=\dfrac{-q}{4\pi r_1^2},~\sigma_{out}=\dfrac{Q}{4\pi r_2^2},\) (Yes, due to Gauss's law) |
| 4. | \(\sigma_{in}=0,~\sigma_{out}=\dfrac{Q+q}{4\pi r_2^2},\) (No, the shape matters) |
a) charge placed at the centre of a shel is +q. Hence, a charge of magnitude -q will be induced to the Surface charge density at the inner surface of the shell is given by the relàtion,
A charge of +q is induced on the outer surface of the shel. A charge of magnitude Q is placed on the outer Surtace of the shell. Therefore, total charge on the outer surface of the shell is Q+ q. Surtace charge density at the outer
(b) Yes
The electric field intensity inside a cavity is zero, even if the shell is not spherical and has any irregular shape.
lake a closed loop such that a part of it is inside the cavity along a field line while the rest is inside the
oNe One aes argeov sopi zero Decause tne neld
inside the conductor IS zero. Hence, electric field is zero, whatever is the shape.
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