Another concentric sphere electric field question

[tony873004]

[SOLVED] another concentric sphere electric field question

Interestingly, the very next question does specify uniformly distributed charges. This inconsistency has me worried that we were not to make that assumption in the 1st question from the other thread. Time to visit the office hours!

Homework Statement

Two concentric plastic spherical shells carry uniformly distributed charges, Q on the inner shell and –Q on the outer shell. Find the electric field (a) inside the smaller shell, (b) between the shells, and (c) outside the larger shell.

The Attempt at a Solution

I imagine the answer for (a) is no field, (b) is same as the previous question, only negative:$$\overrightarrow E = \frac{{ - Q}}{{4\pi r^2 \varepsilon _0 }}{\rm{\hat r}}$$

But I'm not sure about (c). I'm guessing it would be
$$\overrightarrow E = \frac{Q}{{4\pi r_1^2 \varepsilon _0 }}{\rm{\hat r }} - \frac{Q}{{4\pi r_2^2 \varepsilon _0 }}{\rm{\hat r}}$$
where r1 is the distance to the outer shell and r2 is the distance to the inner shell. Is this right? Is there a better way to express this or to simplify this expression?

Perhaps
$$\overrightarrow E = \frac{1}{{4\pi \varepsilon _0 }}\left( {\frac{Q}{{r_1^2 }}{\rm{\hat r }} - \frac{Q}{{r_2^2 }}} \right){\rm{\hat r}}$$

Homework Equations

 

[Shooting Star]

Ans to (a) is correct.

Why is there a minus sign is (b)? The charge on the inner shell is +Q...

In (c), you have goofed up badly. In the other thread, what did you mean by r? I'm leaving it to you to clear the mess up.

 

[tony873004]

Thanks for your reply. I see the point you're making about r. They should be the same because they are the distance from the center of the spheres to a point of interest, rather than the radius of each sphere.

Ok, new attempt. Outside the spheres, they both reduce to a point charge. So they cancel each other. There are no field lines outside the outer sphere.

 

[Shooting Star]

You have got it. [And no minus sign in (b).]