electric field inside a conductor is zero why

Key Concepts

1. Free Electrons in Conductors
   • Metals have plenty of free electrons that can move almost instantly.
2. Electrostatic Equilibrium
   • When the system settles so that charges are no longer moving, we say it is in electrostatic equilibrium.
3. Gauss’s Law Logic

   • For any closed surface completely inside a conductor:

     $\oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0}$

   • Since all excess charge migrates to the outer surface, $q_{enc}=0$. Hence the total flux is $0$, which in a conductor implies $\vec{E}=0$ everywhere inside (not just average zero).

4. Self-Cancellation

   • The moment an external electric field tries to penetrate, free charges rearrange, setting up an internal field $\vec{E}_{internal}$ such that

     $\vec{E}_{external} + \vec{E}_{internal} = 0$

   • Motion stops, so net field inside is zero.

Logical Steps a Student Follows

• Recognize that conductors contain mobile charges.
• Think of the final static condition (no current).
• Apply Gauss’s law to an interior Gaussian surface (find $q_{enc}=0$).
• Conclude: zero flux ⇒ every point inside has $\vec{E}=0$.

Imagine a Big Crowd in a Room

1. Charges Are Like People – Positive and negative charges move around inside a conductor just like people can move around in an empty hall.
2. Someone Starts Pushing (Electric Field) – If an electric field is present, it is like a wind that pushes all people in one direction.
3. People Quickly Rearrange – Those free charges move and pull each other until they settle in new spots (mainly on the surface).
4. No More Push Inside – Once they finish rearranging, they create their own internal push (field) that exactly cancels the external push. So, inside the room, you feel calm—no wind at all.
5. Outside May Still Be Windy – The pushing can stay outside the building (on the surface).

Therefore, inside the conductor, the effective electric field becomes zero.

Step-by-Step Reasoning

1. Take a Gaussian Surface completely inside the bulk of a conductor.

2. Know that all free charges in equilibrium reside on the conductor’s outer surface. So,

   $q_{enc}=0$

3. Apply Gauss’s Law

   $\oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0}=0$

4. Uniform Zero Field

   • The only way the surface integral can be zero for any arbitrary closed surface inside is that the electric field vector itself is zero at every interior point.

5. Conclude

   $\boxed{\vec{E}_{inside\ conductor}=0}$

Hence, the electric field inside a conductor at electrostatic equilibrium is zero.