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Electrostatic Potential and Capacitance — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsElectrostatic Potential and Capacitance3 Marks
Question
A spherical conducting shell of inner radius $r_1$ and outer radius $r_2$ has a charge Q.
  1. A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?
  2. Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain
✓

Answer

  1. Charge placed at the centre of a shell is +q. Hence, a charge of magnitude -q will be induced to the inner surface of the shell. Therefore, total charge on the inner surface of the shell is -q.
Surface charge density at the inner surface of the shell is given by the relation,

$\sigma_1=\frac{\text{Total charge}}{\text{Inner surface area}}=\frac{-\text{q}}{4\pi\text{r}^2_1} \dots\dots(1)$

A charge of +q is induced on the outer surface of the shell. A charqe of magnitude Q is placed on the outer surface of the shell. Therefore, total charge on the outer surface of the shell is Q + q. Surface charge density at the outer surface of the shell,

$\sigma_2=\frac{\text{Total charge}}{\text{Outer surface area}}=\frac{-\text{Q+q}}{4\pi\text{r}^2_2} \dots\dots(2)$
  1. Yes
The electric field intensity inside a cavity is zero, even if the shell is not spherical and has any irregular shape. Take a closed loop such that a part of it is inside the cavity along a field line while the rest is inside the conductor. Net work done by the field in carrying a test charge over a closed loop is zero because the field inside the conductor is zero. Hence, electric field is zero, whatever is the shape.

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