Electric Field in Capacitors

A capacitor is formed of 2 metal plates that are spaced apart:



Here we can see that 2 plates are held apart; one plate is connected to the positive terminal, the other to the negative terminal and an insulator is placed between the plates. Electrons will gather on the negative plate and be repelled from the positive plate, this way the plates become charged and we have an electric field between the two. The voltage across the capacitor will increase until it is equal to the voltage of the supply, at this point it is fully charged and we can say that it has charge stored (on the plates). To calculate the capacitance (the amount of charge that can be stored), we use the equation:

  •  is the Capacitance
  •  is the total charge
  •  is the voltage

And similarly as we found out in the previous tutorial, we can calculate the electric field between the 2 plates by using:

Where  is equal to the total charge () and  is the relative permittivity of the material.

Potential Energy of a Capacitor

Once charged, a capacitor will have a store of electrical potential energy and this can be transferred into electrical energy (and other forms) during discharge. We can calculate this using:

  •  is the energy transfer (work done)
  •  is the total charge
  •  is the voltage
  •  is the Capacitance

Carrier Drift

In Free Space

As we mentioned in the previous tutorial, the force on a point charge in an electric field can be calculated with:

A current carrier (a hole or electron) will accelerate to high velocities if there is a uniform Electric Field present; this is due to the equation shown above along with the equation:

We can then calculate acceleration:   And velocity after time t, when starting at rest (obtained from the SUVAT equations):

Finally we can calculate the distance travelled ():

In a Semiconductor

In a semiconductor, regular collisions will occur due to the surrounding crystal lattice keeping these velocities limited. The average velocity of an electron in an Electric Field can be calculated using:

  •  is average velocity
  •  is the charge
  •  is the Electric Field
  •  is the scattering time, this controls the electrical conductivity of a material
  •  is the mass of the carrier

We usually write the average velocity of charges within a semiconductor as:

  •  is the mobility of the charges.

In a conductor, all of the mobile charge carriers are electrons and they all have the same scattering time. However in a semiconductor (pure or doped), the mobile charge carriers are either electrons or holes and they have different scattering times. For example: Silicon has an electron mobility () of  and has a hole mobility () of . These values are different for each semiconductor material.

Energy of Charges in an Electric Field

As stated before, a charge moving in an electric field will result in an energy change.

Moving Charge (q) in a Uniform Electric Field

Moving Charge (q) in a Uniform Electric Field

In the example above, we have a charge (q) moving through a uniform electric field,  is the changing distance and we can say that the work done in moving the charge by a distance of  is:

There are several important things to note about this equation:

  •  is the dot product and 
  • In this situation
  • It is for these 2 reasons why 

Also that:

  •  is the change in Energy
  •  is the applied Force
  •  is the change in distance
  •  is the charge of the particle
  •  is the electric field strength

In this situation the charged particle is moving against the electric field and so it will be gaining energy, this is shown by  (if the particle was moving in the opposite direction, the left side of the equation would be negative and so the particle would be losing energy).

The total work done (moving from point A to point B) can be calculated by integrating:

By the law of conservation of energy, this change in energy will equal the difference between the 2 potentials in the electric field:

  •  is the potential energy

Knowing this energy, we can then go onto calculate the potential difference:

Note: this is essentially the same calculation as in the previous tutorial (Calculating Electric Potential).

Calculating Capacitance of a Capacitor

We learned at the start of this tutorial that:

And that in a uniform electric field:

Therefore it can be shown that:

Move onto the final tutorial where we’ll learn about the current in semiconductors and about Conductivity and Resistivity.