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- Homework Statement:
- I have attached a pdf file of the entire problem. I am concerned with problem 23.03.

- Relevant Equations:
- ##\epsilon_0 \oint \vec{E}\cdot d\vec{A} = q_{enc}##

Attached is problem 23.03 from Halliday and Resnick.

We have a sphere of uniform negative charge Q = -16e and radius R = 10cm. at the center of the sphere is a positively charged particle with charge q = +5e. We are supposed to use Gauss' law to find the magnitude of the electric field at point ##P_1## a distance r = 6cm from the center of the sphere. So we construct a sphere through the point ##P_1## and try to find the field. The outer charged sphere and the point charge both are symmetric, the field will always perpendicular to the sphere we constructed. Then we have

##\epsilon_0 E \oint dA = \epsilon_0 E (4\pi r^2) = q_{enc} \implies E = \cfrac{1}{4\pi \epsilon_0}\cfrac{q_{enc}}{r^2}##

But this is the same as if we just used coulomb's law to find the field of the single point charge (the one at the center) at ##P_1##. Why is that? The book doesn't explain this weird result. What happened to the field set up by the charged sphere?

It also doesn't tell us anything about the direction of E, just the magnitude. How can we know the direction of the field? Or was the direction necessary to set up the solution and I just didn't understand it?

We have a sphere of uniform negative charge Q = -16e and radius R = 10cm. at the center of the sphere is a positively charged particle with charge q = +5e. We are supposed to use Gauss' law to find the magnitude of the electric field at point ##P_1## a distance r = 6cm from the center of the sphere. So we construct a sphere through the point ##P_1## and try to find the field. The outer charged sphere and the point charge both are symmetric, the field will always perpendicular to the sphere we constructed. Then we have

##\epsilon_0 E \oint dA = \epsilon_0 E (4\pi r^2) = q_{enc} \implies E = \cfrac{1}{4\pi \epsilon_0}\cfrac{q_{enc}}{r^2}##

But this is the same as if we just used coulomb's law to find the field of the single point charge (the one at the center) at ##P_1##. Why is that? The book doesn't explain this weird result. What happened to the field set up by the charged sphere?

It also doesn't tell us anything about the direction of E, just the magnitude. How can we know the direction of the field? Or was the direction necessary to set up the solution and I just didn't understand it?