Max voltage of an air capacitor

[HelloCthulhu]

Homework Statement

(a) A parallel plate capacitor has area A = 1 cm², a plate separation of d = 0.01m, and is filled with air. If the breakdown field is E₀ = 3 × 10⁶V/m, calculate the maximum voltage and charge the capacitor can hold.

Homework Equations

dielectric constant of air = 1

Q=CV

V_max = E₀ x d

The Attempt at a Solution

The breakdown field is the minimum electric field at which a material ionizes; in air, this means a sparkforms.Maximum voltage:$$V{max}=\varepsilon_0\times d =(3\times10^6\times0.01) =(3\times10^4)V = 30kV$$To find the maximum charge corresponding to V = 30kV, we need the capacitance of the parallel platecapacitor,$$\frac{(8.85×10^{-12})\times(0.0001)}{0.01}=\frac{(8.85×10^-16)}{0.01}=8.85×10^{-14}F=0.0885pF$$$$V{max}=(3\times10^4 )\times(0.0885\times10^{-12}) =(3\times10^4)V=30kV$$The charge is determined from C = Q/V :$$Q=CV=(3\times10^4 )\times(0.0885\times10^{-12}) =2.655\times10^9C=2.655nC$$

 

[kuruman]

The charge is determined from C = Q/V

$$\frac{(8.85×10^{-12})\times(0.0001)}{0.01}=\frac{(8.85×10^-16)}{0.01}=8.85×10^{-14}F=0.0885pF$$
$Q=CV=(3\times10^4 )\times(0.0885\times10^{-12}) =2.655\times10^9C=2.655nC$

Why two different values for the capacitance?

 

[HelloCthulhu]

Why two different values for the capacitance?

I might not understand what you mean by "two different values for the capacitance", but I'll try to clarify.

To find the maximum charge for the capacitor, I first have to find the capacitance:

$$\frac{(8.85×10^{-12})\times(0.0001)}{0.01}=\frac{(8.85×10^-16)}{0.01}=8.85×10^{-14}F=0.0885pF$$

This is the maximum charge:

$$Q=CV=(3\times10^4 )\times(0.0885\times10^{-12}) =2.655\times10^9C=2.655nC$$

 

[kuruman]

No problem, my mistake. I looked at the powers of 10 not noticing that you shifted the decimal over. It looks fine except that 10⁹C should be 10^-9C, but I know what you mean because you have nC in the end.

 

[HelloCthulhu]

Oops! Thank you for catching that!

Maximum charge:

$$Q=CV=(3\times10^4 )\times(0.0885\times10^{-12}) =2.655\times10^-9C=2.655nC$$