- #1
tshafer
- 42
- 0
I'm calculating the capacitance of a set of spherical shells of radii b > a. To do this, I place a charge +Q on the inner shell and -Q on the outer shell so that I get the electric field vector pointing outward
[tex]
\vec E = \frac{Q}{4\pi\varepsilon_0} \frac{\hat r}{r^2}.
[/tex]
Finding the potential should be trivial... I do the integral
[tex]
V = -\int_a^b \vec E \cdot d\vec r = -\frac{Q}{4\pi\varepsilon_0} \int_a^b \frac{dr}{r^2} = -\frac{Q}{4\pi\varepsilon_0}\left( \frac{1}{a} - \frac{1}{b} \right).
[/tex]
This is off by a minus sign and I cannot understand why... reversing the limits of integrations, of course, kills the sign but introduces a sign flip in the dot product as E then opposes dr. This should be trivial... help? :)
[tex]
\vec E = \frac{Q}{4\pi\varepsilon_0} \frac{\hat r}{r^2}.
[/tex]
Finding the potential should be trivial... I do the integral
[tex]
V = -\int_a^b \vec E \cdot d\vec r = -\frac{Q}{4\pi\varepsilon_0} \int_a^b \frac{dr}{r^2} = -\frac{Q}{4\pi\varepsilon_0}\left( \frac{1}{a} - \frac{1}{b} \right).
[/tex]
This is off by a minus sign and I cannot understand why... reversing the limits of integrations, of course, kills the sign but introduces a sign flip in the dot product as E then opposes dr. This should be trivial... help? :)