Question
Why is the electric field inside a conductor zero in electrostatic equilibrium? Prove this using Gauss’s law.
This is a NCERT Class 12 favourite and shows up in JEE Main almost every year — sometimes as a direct question, sometimes hidden inside a problem about shielding or induced charges.
Solution — Step by Step
A conductor has free electrons that can move. The moment any external electric field is applied, these electrons experience a force and start migrating. They keep moving until the situation is stable — that stable state is called electrostatic equilibrium.
Say an external field points to the right inside the conductor. Electrons (negative) drift left, positive charge builds up on the right face. This separation creates an internal field pointing left — opposing the external field. Electrons keep shifting until exactly. At that point, the net field inside is zero and the charges stop moving.
Draw a Gaussian surface — any closed surface — entirely inside the conductor. Since charges are only on the surface of a conductor (they repel each other outward to the boundary), there is no free charge enclosed by this Gaussian surface.
Gauss’s law says:
With :
This means at every point inside. (If were non-zero somewhere, the flux through a surface enclosing that point couldn’t be zero — contradiction.)
Since inside, and Gauss’s law with any interior Gaussian surface gives , we conclude: all excess charge on a conductor resides only on its outer surface. This is a direct corollary and frequently asked separately in boards.
Why This Works
The argument has two layers. The physical layer: free electrons in a conductor are genuinely free — they respond to any net field until they’ve redistributed enough to cancel it out. Nature forces the system to equilibrium because any net field would drive a current, and currents imply energy dissipation, which can’t sustain itself forever.
The mathematical layer: Gauss’s law connects flux to enclosed charge. Once we know no charge can sit inside (because if it did, there’d be a field, which would push that charge to the surface anyway), the flux is zero, so the field must be zero. The two arguments reinforce each other perfectly.
This is also why conductors act as electrostatic shields. Faraday cage works on exactly this principle — sensitive equipment inside a conducting enclosure sees zero external electric field regardless of what’s happening outside.
Alternative Method — Energy Argument
This approach is cleaner for conceptual MCQs in JEE Main.
If there were a non-zero field inside the conductor, free electrons would experience a force and accelerate. Accelerating charges imply kinetic energy gain, which must come from somewhere. In a static situation, there’s no energy source driving this — so the system would violate energy conservation unless the electrons are already stationary. Electrons are stationary only when .
This isn’t a replacement for the Gauss’s law proof (which is more rigorous), but it gives you an instant physical intuition for MCQs.
Common Mistake
Students often confuse inside the conductor with inside a cavity in the conductor. The field is zero inside the conducting material itself. If there’s a hollow cavity with no charge inside it, the field there is also zero — but for a slightly different reason (superposition of surface charge contributions). If there’s a charge inside the cavity, the field inside the cavity is NOT zero. Many students blanket-apply “field inside conductor = zero” to the cavity as well, which is wrong when charges are present inside the cavity.
One more trap in boards: the question sometimes asks about the field “just outside” a conductor. That is not zero — it equals (where is surface charge density). The field jumps from zero to as you cross the surface. This discontinuity itself is a standard 2-mark derivation in CBSE Class 12.