To check the total induced charge (which had better equal the value of the image charge) we need to integrate the charge density over the whole spherical surface: The total induced charge (remember this is a sphere, P is on a circle of equal induced charge, shown dashed)

• There is negative surface charge density on the metal surface • The resulting outside field is due to both the point charge and the induced surface charge density on the metal

The induced charge distribution is negative and the charge density is greatest at (x = 0, y = 0, z = 0). The total charge on the conductor can be calculated by surface integrating of σ:

Thus for the determination of the surface charge on the conductor, you have to take Griffiths equation (#) $$\sigma=-\epsilon_0 \frac{\partial V}{\partial n}$$ with the normal derivative of this potential solution at $z \to +0$.

- We can use this to find the induced surface charge density on the sphere. 3. Surface Charge Density - Start with the general form found previously: [ E2−E1 ⋅n= 1 0 ] n= 0 - For a conductor, the electric field inside is everywhere zero, E1 = 0: [E2⋅n= 1 0 ] n=n 0