Spherical capacitor with 2 dielectrics.

[Lavabug]

Homework Statement

The problem is on page 40 of this PDF:
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide05.pdf
Find the capacitance of a spherical capacitor with 2 different homogeneous dielectrics arranged concentrically.

The Attempt at a Solution

In that document they simply apply the spherical capacitance formula and get the expressions for two capacitors, the inner and outer sphere, then they add their inverses as its pretty much 2 capacitors in series.

I've been trying to solve the problem "my way" by starting straight from the fact that E is the field for a sphere of charge, and calculate the voltage: circulation integral from the surface of the inner terminal to the outer shell/terminal.

So I have to add the following integrals

$V = \int^a_b Q/4\pi\epsilon_1r^2\,dr + \int^b_c Q/4\pi\epsilon_2r^2\,dr$

Is this correct? Because if I do the math/LCM and divide Q by it etc, my expression for the capacitance differs. Am I assuming something invalid with my integral? I know E should be constant between the cap terminals, but the field lines circulate through 2 different dielectrics so I thought splitting up the integral was reasonable.

EDIT: OOPS nevermind, lack of simplifying my answer made me think it was wrong when I had arrived to the same answer as that pdf.

Just want to make sure in case though, did I calculate V correctly?

 

[tiny-tim]

Hi Lavabug!

Yes, your V for a spherical https://www.physicsforums.com/library.php?do=view_item&itemid=112" is fine. ​