Table of contents
Here we want to derive electric potential, electric field and capacitance.
Electric potential in plate capacitor
In the following, the electric potential \(\varphi\) between two capacitor plates is derived. The plate area is \(A\) and the plates are at a distance \(d\) from each other.
For the derivation of the potential, Poisson's equation is used:
Since there are no charges between the plates, the charge density \(\rho\) (charge per volume) inside is zero. The Poisson equation simplifies to the Laplace equation:
The three-dimensional Laplace equation
2 can be reduced to the one-dimensional case because the plates are homogeneously charged and the potential \(\varphi\) is thus independent of the \(y\) and \(z\) coordinates:
Thus, to find the electrostatic potential \(\varphi\) inside the plate capacitor, the differential equation
3 must be solved. However, this is quite simple, because the second spatial derivative which is zero corresponds to a linear function:
Here \(a\) and \(b\) are constants. That this form of \(\varphi\) must be correct can easily be checked by differentiating twice with respect to \(x\). It yields zero, as required by Laplace's equation
Now the slope \(a\) and the y-axis intercept \(b\) of the function \(\varphi(x)\) must be determined, because they are still unknown. For this purpose the boundary conditions of the problem are used.
Boundary condition #1: The first capacitor plate is placed at \(x=0\) and has the constant potential \(\varphi_1\) there.
Boundary condition #2: The second capacitor plate is placed at \(x=d\) and has the constant potential \(\varphi_2\) there.
Now the first boundary condition is substituted into
Thus \(b\) is now determined. Here \(b\) represents the potential at the first plate. Now the second boundary condition must be used. Substitute it into
6 \(b\) occurs. But we have already determined \(b\) in Eq.
But the potentials \(\varphi_1\) and \(\varphi_2\) are not known either. What is known, however, is the applied voltage \(U\) between the capacitor plates! It is given by the potential difference:
7 for \(\varphi_1 - \varphi_2\) and insert voltage
The voltage \(U\) and the distance \(d\) of the plates are known and therefore the slope is also known:
Now the determined constants
10 have to be inserted into the potential equation
4 to get the potential inside the capacitor:
Electric field in plate capacitor
To express the electric field using the known voltage \(U\), the spatial derivative of the potential (gradient equation) is used (in the one-dimensional case):
Differentiating the previously determined potential
11 gives the electric field inside the capacitor:
Capacitance of the plate capacitor
In the following, the capacitance \(C\) of the plate capacitor is derived, which tells us how good the plate capacitor can 'store' the electric charge.
The electric field \(E\) of a charged plate is given by:
Here \(\sigma = Q/A\) represents the surface charge density of the charged plate. So \( \sigma \) is charge \(Q\) per plate area \(A\):
The electric field is proportional to the charge according to
15. And since the electric field is also proportional to the voltage according to
13, the voltage is proportional to the charge. The constant of proportionality \(C\) is called the electric capacitance:
In the case of a plate capacitor, the charge \(Q\) on the plate is unknown; the capacitance \(C\) is also unknown. Only the voltage \(U\) is given by a voltage source and is thus known. So the goal is to figure out the charge of the capacitor in order to be able to calculate the remaining unknown, namely the capacitance \(C\).
13 to have an equation for \(Q\) that contains only known quantities. Then rearrange the resulting equation for \(Q\):
Now just insert
16 and rearrange for the capacitance: