7.4.2 Parallel plate capacitor

1. Properties of Dielectrics:

• Dielectrics are materials that can be polarised when an electric field is applied.
• Dielectrics are characterised by a property called permittivity (ε), which indicates how well they can store an electric field within them.
• The relative permittivity (ε_r), also known as the dielectric constant, is the ratio of the permittivity of the dielectric to the permittivity of free space $(ε_0)$. It is given by:

$$ε_r = \frac{ε}{ε_0}$$

2. Capacitance of a Parallel Plate Capacitor:

• The capacitance ($C$) of a parallel plate capacitor depends on three factors:
• Area of the plates ($A$): Larger plate areas result in higher capacitance, as there is more space to store charge.
• Distance between the plates ($d$): A smaller gap increases the capacitance, as the electric field strength is higher.
• Relative permittivity (ε_r): A higher relative permittivity of the dielectric increases the capacitance, as it reduces the field strength required for a given amount of charge. The capacitance formula is:

$$C = \frac{Aε_0ε_r}{d}$$

3. Polarisation of the Dielectric:

• Dielectrics are made up of polar molecules, which have one positive and one negative end. In the absence of an electric field, these molecules are randomly aligned.
• When an electric field is applied, the polar molecules align with the field. The positive ends face the negatively charged plate, and the negative ends face the positively charged plate, as shown in the diagram.
• Each molecule in the dielectric develops its own internal electric field that opposes the external field created by the capacitor. This effect reduces the overall electric field in the gap between the plates.
• As a result, less potential difference is needed to store a given amount of charge, so capacitance increases.

4. Impact on Capacitance:

• Since the required potential difference to achieve the same charge decreases, the capacitance increases. Using the relationship:

$$C = \frac{Q}{V}$$

we see that a lower $V$ (for a given $Q$) results in a higher $C$.

Key Formulae

• Relative Permittivity:

$$ε_r = \frac{ε}{ε_0}$$

• Capacitance of a Parallel Plate Capacitor:

$$C = \frac{Aε_0ε_r}{d}$$

Diagram Summary

• The diagram shows a parallel plate capacitor with:
  • Positive and negative charges on opposite plates.
  • A dielectric material between the plates.
  • An electric field $E$ directed from the positive to the negative plate.
  • Polar molecules within the dielectric aligned with the field, reducing the net field strength.