Parallel Plate Capacitor

How does a parallel plate capacitor work?

The electrical charges of the material are separated proportionally to the electrical field, creating two poles, a negative and a positive one.

The electric polarisation process is similar to magnetisation, where a magnetic dipole is induced in a magnetic material when placed near a magnet.

Therefore, when dielectric materials are placed in an external electrical field, the dipole moment that is induced per unit volume of the dielectric material is also known as electric polarisation. This is described by the equation below, where k is the dimensionless dielectric constant, E the permittivity of the material, and Eo the permittivity of vacuum, which is around 8.85 × 10^-12 farad per metre (F/m).

$k=\frac{E}{E_0}$$k=\frac{E}{E_0}$

The two plates of the parallel plate capacitor are connected to a power supply. The plate that is connected to the positive terminal of the battery acquires a positive charge, while the plate that is connected to the negative terminal acquires a negative charge. This happens because the positive pole pushes electrons to the opposite plate. Due to the attraction between the positive and negative charges acquired in the positive and negative plates, the charges are stored within the plates of the capacitor.

Electric field lines are formed between the two plates from the positive to the negative charges, as shown in figure 1. The polarisation of the dielectric material of the plates by the applied electric field increases the capacitor’s surface charge proportionally to the electric field strength in which it is placed.

Figure 1. Parallel plate capacitor configuration. Source: toppr.com.

Parallel plate capacitor: Derivation

The two plates of a parallel plate capacitor are separated by a distance d measured in m, which is filled with atmospheric air. The cross-sectional area of each plate A is measured in m². The electric field E of each plate is equal to the following, where σ is the surface density.

$$\begin{gathered} E=\frac{\sigma }{2·{\epsilon }_0} \\ \sigma =\frac{q}{A}\Rightarrow E=\frac{q}{A·{\epsilon }_0} \end{gathered}$$

If the potential difference between the two plates is equal to V, when we substitute the equation found for the electric potential, we get:

$$\begin{gathered} E=\frac{V}{d}\Rightarrow V=E·d \\ V=d·\frac{q}{{\epsilon }_0·{\rm A}} \end{gathered}$$

Now, substituting the capacitance in the derived voltage, we get:

$$\begin{gathered} C=\frac{q}{V}=\frac{q·{\epsilon }_0·A}{q·d} \\ C\left[F\right]=\frac{{\epsilon }_0·A}{d} \end{gathered}$$

It can be seen that the capacitance depends on the distance between the plates. The charge stored is proportional to the surface area and inversely proportional to distance. This can also be validated by considering the characteristics of the Coulomb force, where like charges repel and unlike charges attract each other. The force between charges decreases with distance. The bigger the plates, the greater the charge storing capacity as the charges spread out more. Thus, the storable charge is increased when the area is also increased. Similarly, the closer the plates, the greater the attraction force between the opposite charges, so capacitance should be greater when the distance is decreased.

Parallel Plate Capacitor: Leakage currents

A given charge is supplied to each plate. Because there is no ideal dielectric material that can hold the charge perfectly, the increase in the potential leads to leakage currents, which cause the capacitor to discharge in an unwanted way once it is disconnected from the circuit.

Parallel plate capacitor: Electric field

In a parallel plate capacitor, when a voltage is applied between two conductive plates, a uniform electric field between the plates is created. However, at the edges of the two parallel plates, instead of being parallel and uniform, the electric field lines are slightly bent upwards due to the geometry of the plates. This is known as the fringing or edge effect (see figure 2).

A capacitor’s electric field strength is directly proportional to the voltage applied while being inversely proportional to the distance between the plates.

Figure 2. Diagram showing the fringing of the electric field at the edges of the two plates.

Usage of parallel plate capacitors

The usage of capacitors range from filtering static out of radio reception to energy storage in heart defibrillators and include the following:

• Energy storage and circuit protection against unusual spikes in voltage or any interruption in the circuit.
• Electronic signal processing.
• Suppression and coupling.
• Motor starters used in pumps and compressors such as refrigerator compressors.

A squared length capacitor is a capacitor that has the same width and length. Hence, its area can be calculated by the squared length.