Question

A conductor with a cavity is charged positively and its surface charge density is $\sigma$. If $E$ and $V$ represent electric field and potential, then inside the cavity

• A. $\sigma = 0$ and $V = 0$
• B. $E = 0$ and $V = 0$
• C. $E = 0$ and $\sigma = \text{constant}$
• D. $V = 0$ and $\sigma = \text{constant}$
• E. $E = 0$ and $V = \text{constant}$

Hint:

Inside any conductor (and empty cavity inside it), electric field is zero and potential is uniform.

Answer 1

Answer: (E)

Concept: In electrostatics, a conductor in equilibrium has the following fundamental properties:
• Electric field inside a conductor is zero
• Entire conductor (including cavity) is an equipotential body
• Excess charge resides only on the outer surface

Step 1: Behaviour of charges in conductor
When a conductor is charged:
• Free electrons rearrange themselves
• They move until internal electric field becomes zero If any electric field existed inside:
• Charges would keep moving
• Equilibrium would not be achieved Hence, \[ E_{\text{inside conductor}} = 0 \]

Step 2: Field inside cavity
Since the cavity contains no charge:
• No field lines originate or terminate inside
• Outer surface charges rearrange to cancel any internal field Thus, \[ E_{\text{inside cavity}} = 0 \]

Step 3: Potential inside conductor
Electric field is related to potential by: \[ E = -\frac{dV}{dx} \] If $E = 0$, then: \[ \frac{dV}{dx} = 0 \Rightarrow V = \text{constant} \] Hence, entire conductor (including cavity) is at same potential.

Step 4: Why other options are wrong:
• $\sigma = 0$ → incorrect (charge exists on surface)
• $V = 0$ → not necessarily zero, only constant
• $\sigma = \text{constant}$ inside cavity → meaningless (no surface there) Final Conclusion: \[ E = 0, \quad V = \text{constant} \]