Field at 45° from charged sphere and plane with σ₀ = ρ₀R/2

[matpo39]

ok i was flying through the homework no prob. then i hit this problem and i got an answer but i don't think its right.

A uniformly charged sphere of radius R and volume charge density $$\rho_0$$ is adjacent to a uniformly charged infinite plane of surface charge density $$\sigma_0$$. the charge densities are related by
$$\sigma_0=\frac{\rho_0R}{2}$$
the center of the sphere is a distance d from the plane. Find two points, one inside the sphere and one outside the sphere where the electric field is oriented away from the plane at a 45 degree angle with respect to the z axis.[note these points are not on the axis] (in the figure the infinite plane lies in the xy plane )

well i started this off by finding the electric field inside the sphere

$$\vec{E_s}=\frac{\rho_0R}{3\epsilon_0}$$

i then found the charge of the infinite plane via the pill box gaussian surface and came up with
$$\vec{E_p}=\frac{\sigma_0}{2\epsioln_0}=\frac{\rho_0R}{4\epsilon_0}$$

$$\vec{E_s}+\vec{E_p}=\frac{7\rho_0R}{12\epsilon_0}$$

and breaking to components i got
$$\frac{7\rho_0R}{12\epsilon_0}(cos45+sin45)$$

and by a similar approch i got
$$[\frac{\rho_0R}{\epsilon_0}(\frac{R^2}{3r^2}+\frac{1}{4})](cos45+sin45)$$

like i said i don't think this is right, so if some one could help me out a bit that would be great

thanks

 

[quasar987]

Mmmh. There's a problem here: you write $\vec{E}= \mbox{a scalar}$. What are the directions of the E_P and E_S vectors? Write them in cartesian coordinate for a coordinate system centered on the sphere, and find the condition on E_p + E_s to be at 45°.

Oh, and your field equations look wrong. The field inside the sphere is not uniform.