Capacitance between two coplanar electrodes

[garydia1]

Homework Statement

I have two coplanar electrodes (180 deg between them) and one of them is under voltage causing circular electric field between them. The target is to calculate the capacitance.

Homework Equations

Using the capacitance equation C = $$\frac{\epsilon_{0}A}{l}$$
I say that A = bdx and l = $$\pi$$r
b - is out of plane length of the electrodes

The Attempt at a Solution

I'm integrating the equation from R1 to R2 (coordinates of electrode on x axis)
but I have to substitute dx with something else - x = rCos$$\theta$$ so that
dx = cos($$\theta$$)dr

so the solution is C= $$\frac{\epsilon_{0}bCos\theta}{\pi}ln\left(\frac{R2}{R1}\right)$$

Am I doing it rigth?

 

[anathema]

Your approach to calculating the capacitance is correct. However, there are a few things that can be improved in your solution.

Firstly, when integrating from R1 to R2, the variable of integration should be r, not x. This is because the equation for the capacitance uses r as the distance between the electrodes, not x.

Secondly, the equation for the capacitance is C = \frac{\epsilon_{0}A}{d}, where d is the distance between the electrodes. In your solution, you have used b as the distance between the electrodes. This may be a typo, but it's important to use the correct distance in the equation.

Finally, your solution does not take into account the fact that the electric field is circular. The electric field is not constant throughout the entire region between the electrodes, so you cannot simply use the formula for a parallel plate capacitor. Instead, you will need to integrate the electric field over the entire region to get an accurate value for the capacitance.

Overall, your approach is correct, but there are a few adjustments that need to be made for a more accurate solution. Keep up the good work!