The charge density σ\sigma σ on the surface of a conductor at a point where the electric field just outside the surface has magnitude EEE and is directed toward the conductor is given by:
σ=−ϵ0E\sigma =-\epsilon_0 Eσ=−ϵ0E
Here, ϵ0\epsilon_0 ϵ0 is the permittivity of free space.
Explanation:
- The electric field just outside a conductor's surface is related to the surface charge density by Gauss's law.
- For a conductor, the electric field inside is zero, so the discontinuity of the electric field across the surface is due to the surface charge.
- If the electric field points toward the conductor (inward), the surface charge must be negative, hence the negative sign.
- Using a Gaussian pillbox that straddles the surface, the relation between the electric field and surface charge density is:
Eoutside−Einside=σϵ0E_{\text{outside}}-E_{\text{inside}}=\frac{\sigma}{\epsilon_0}Eoutside−Einside=ϵ0σ
Since Einside=0E_{\text{inside}}=0Einside=0, we get:
E=σϵ0 ⟹ σ=ϵ0EE=\frac{\sigma}{\epsilon_0}\implies \sigma =\epsilon_0 EE=ϵ0σ⟹σ=ϵ0E
But because the field is directed toward the surface, σ\sigma σ is negative:
σ=−ϵ0E\sigma =-\epsilon_0 Eσ=−ϵ0E
This matches the sign convention that positive surface charge produces an outward field, so inward field corresponds to negative surface charge
Summary:
Quantity| Expression| Notes
---|---|---
Electric field outside conductor| EEE (magnitude)| Directed toward surface
Surface charge density σ\sigma σ| σ=−ϵ0E\sigma =-\epsilon_0 Eσ=−ϵ0E| Negative
sign due to inward field
This formula directly relates the electric field magnitude just outside the conductor to the surface charge density at that point.
