Finding Electric Field of a Uniformly Charged Sphere & Plane

[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.

 

[quicknote]

Hello, thank you for sharing your approach to solving this problem. It seems like you are on the right track, but there are a few things that need to be clarified.

Firstly, when finding the electric field inside the sphere, you need to take into account the distance from the center of the sphere to the point where the electric field is being measured. This should be included in your equation for \vec{E_s}.

Secondly, when adding the electric fields of the sphere and the plane, you need to consider the direction of the electric fields. Since the electric field of the plane is perpendicular to the surface, it should not affect the direction of the electric field at the points you are looking for. Therefore, the electric field at these points should only be influenced by the electric field of the sphere.

Lastly, when breaking the electric field into components, you should use the cosine and sine functions for the angle between the electric field vector and the x and y axes, respectively. In this case, since the electric field is at a 45 degree angle with respect to the z axis, the angle between the electric field vector and the x and y axes is also 45 degrees.

Overall, it seems like you have a good understanding of the concepts involved in finding the electric field of a uniformly charged sphere and plane. I would suggest reviewing your calculations and making sure you are taking into account all the necessary factors. If you are still unsure, I would recommend seeking help from a teacher or classmate. Good luck!