Derive formulas of electric field & potential difference between charged plates

Last updated on April 15th, 2022 at 10:35 am

Here we will derive the formulas of electric field & potential difference between charged plates.

fundamentals

Now, let’s see some of the facts we need to know (or revise) first, before going for the derivation.

1 ) The electric field strength (E) surrounding a point charge Q using the following equations:
|E| = |F_e| /q
=> |E| = kQ/r² ………….. (1)

2 ) The electric field around a point charge is a non-uniform electric field. Its magnitude depends on the distance from the charge.

3 ) A special type of electric field exists between two charged parallel plates. The magnitude of the electric field between the plates is uniform anywhere between the plates and it can be determined using the general equation for an electric field, |E| = |F_e| /q.

But here we can’t use equation number 1 (given above as |E| = kQ/r²) to represent the electric field between the parallel plates, because the electric field equation in equation 1 is used only for point charges.

Now, you can see how another equation for determining the electric field strength between plates arises from an important relationship between the uniform electric field and the electric potential difference between two charged parallel plates.

Derivation of formulas of electric field & potential difference between charged plates

If a small positively charged particle (q) is moved through the uniform electric field (E), a force is required, where F= Eq. This force is the force exerted on the particle due to the presence of the electric field. If this force moves the charged particle a distance (d) between the plates, then the work done is:
W= |F|d
=> W = |E| q d

Since this system is conservative, the work done is stored in the charge as electric potential energy:

W = E_p = |E| q d

The electric potential difference between the parallel charged plates is:
V = electric potential energy/q = |E| q d / q = |E| d
=> V = |E| d [derived]

To calculate the magnitude of the uniform electric field between charged plates, use the equation:
|E|=V/ d, [derived]
where V is the electric potential difference between two charged plates in volts; d is the distance in meters between the plates; and |E| is the magnitude of the electric field in volts per meter.

Note that 1 V/m equals 1 N/C because 1 V/m = 1 J/C /1 m = 1 Nm/C/ 1 m = 1 N/C

Summary

1) The magnitude of the electric field between the plates is uniform anywhere between the plates and it can be determined using the general equation for an electric field, |E| = |F_e| /q.

2) We can’t use equation number 1 (given above as |E| = kQ/r²) to represent the electric field between the parallel plates, because the electric field equation in equation 1 is used only for point charges.

3 ) The formula of the electric potential difference between the parallel charged plates is:
=> V = |E| d [derived]

4 ) The formula of the magnitude of the uniform electric field between charged plates (in terms of electric potential difference V) is:
|E|=V/ d [derived]