Maximum charge on a spherical capacitor

[lorenz0]

Homework Statement

A spherical capacitor has internal radius $R_1$ and external radius $R_2$.
Between the spheres there is a dielectric with constant $\varepsilon_r$.
If the maximum electric field that can be applied without electrical discharges occurring is $E_{max}$, find the corresponding maximum charge that can be put on the plates.

Relevant Equations

$\Delta V=\int \vec{E}\cdot d\vec{l}$

The electric field is the one generated by the charge $+Q$ on the inner sphere of the capacitor, which generates a radial electric field $\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}$ which, due to the presence of the dielectric, become $\vec{E}_d=\frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{Q}{r^2}\hat{r}$ so $\Delta V=\int \vec{E}_d\cdot d\vec{l}=\int_{R_1}^{R_2}\frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{Q}{r^2}\hat{r}\cdot d\vec{l}=\frac{Q}{4\pi\varepsilon_0\varepsilon_r}\frac{R_2-R_1}{R_1R_2}.$

So, $E=\frac{\Delta V}{R_2-R_1}=\frac{Q}{4\pi\varepsilon_0\varepsilon_r}\frac{1}{R_1R_2}\leq E_{max}\Rightarrow \frac{Q_{max}}{4\pi\varepsilon_0\varepsilon_r}\frac{1}{R_1R_2}= E_{max}\Leftrightarrow Q_{max}=E_{max}4\pi\varepsilon_0\varepsilon_rR_1R_2$.

Does this make sense? Thanks

 

[hutchphd]

Why are you using the potential at all? The limit is given in terms of the E field.
Where is it biggest?

 

[lorenz0]

Why are you using the potential at all? The limit is given in terms of the E field.
Where is it biggest?

$E$ is biggest on the surface of the inner sphere so $E_{max}=E(R_1)=\frac{Q_{max}}{4\pi\varepsilon_0\varepsilon_r R_1^2}\Leftrightarrow Q_{max}=4\pi\varepsilon_0\varepsilon_r R_1^2 E_{max}$. Is this correct?

 

[hutchphd]

Yes.

 

[lorenz0]

t

Yes.

Thank you.