assume that at some point just outside the surface of the conductor, the electric field has magnitude e and is directed toward the surface of the conductor. what is the charge density σ on the surface of the conductor at that point?
The surface charge density is
σ=− ε0 E\sigma =-,\varepsilon_0,Eσ=−ε0E
at that point on the conductor. Here’s why, in words:
- For a conductor in electrostatic equilibrium, the electric field just outside the surface is related to the local surface charge density by
E=σε0E=\frac{\sigma}{\varepsilon_0}E=ε0σ
and the field is directed perpendicular to the surface.
- By convention, the electric field direction is the direction a positive test charge would accelerate, i.e., away from positive charge and toward negative charge.
- If the field at the surface is directed toward the conductor, that means the nearby surface charge must be negative (it attracts a positive test charge).
- Therefore σ\sigma σ is negative and has magnitude ε0E\varepsilon_0 Eε0E, giving
σ=− ε0 E.\sigma =-,\varepsilon_0,E.σ=−ε0E.