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=−ϵ0​E

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​=ϵ0​σ​

Since Einside=0E_{\text{inside}}=0Einside​=0, we get:

E=σϵ0 ⟹ σ=ϵ0EE=\frac{\sigma}{\epsilon_0}\implies \sigma =\epsilon_0 EE=ϵ0​σ​⟹σ=ϵ0​E

But because the field is directed toward the surface, σ\sigma σ is negative:

σ=−ϵ0E\sigma =-\epsilon_0 Eσ=−ϵ0​E

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=−ϵ0​E| 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.