Question

If you use a parallel plate capacitor with air in the gap between the plates as part of a series RLC circuit in a generator, you can measure current flowing through the generator. Why is it that the air gap in the capacitor does not act like an open switch, blocking all current flow in the circuit?

Short answer

Answer: In an AC circuit, the current does not need a continuous conductive path, as it alternates its direction. Capacitors in an RLC circuit allow charges to accumulate and discharge during positive and negative half cycles of the alternating current, enabling continuous current flow. The air gap acts as a dielectric medium, allowing the electric field to establish a potential difference between the capacitor plates. This causes an alternating current to flow through the circuit even in the presence of an air gap. Thus, the air gap does not open the circuit or act like an open switch.

Step-by-step solution

Understanding a parallel plate capacitor with an air gap

A parallel plate capacitor is an arrangement of two parallel conductive plates separated by an insulating material called a dielectric. When a voltage is applied across the plates, the electric field is created, which allows the charges to accumulate on the plates. In this case, air is acting as the dielectric medium between the capacitor plates.

Understanding a series RLC circuit

A series RLC circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series with a generator (AC voltage source). The impedance of the circuit is essential for understanding the behavior of the current flow in the circuit. The impedance of the RLC circuit, Z, is a complex quantity given by: $$Z = R + j(X_L - X_C)$$ Where $$X_L$$ is the inductive reactance, $$X_C$$ is the capacitive reactance, and $$j$$ is the imaginary unit.

Capacitive reactance and air gap

Capacitive reactance is a measure of a capacitor's opposition to the AC current flow, given by the formula: $$X_C = \dfrac{-1}{2 \pi fC}$$ Where $$f$$ is the frequency of the AC voltage source, and $$C$$ is the capacitance of the capacitor. The capacitance of a parallel plate capacitor with an air gap (having relative permittivity $$\varepsilon_r$$) is given by: $$C = \dfrac{\varepsilon_0 \varepsilon_r A}{d}$$ Where $$\varepsilon_0$$ is the vacuum permittivity, $$A$$ is the area of the plates, and $$d$$ is the distance between the plates (the air gap). As per the equation, the capacitance will decrease with an increase in the air gap ($$d$$), which will increase the capacitive reactance ($$X_C$$) and affect the overall impedance ($$Z$$) of the RLC circuit.

Why air gap does not block the current

In an AC circuit, the current does not need continuous conductive path, as it continually alternates its direction. Capacitors in RLC circuit allows for charges to accumulate and discharge during positive and negative half cycles of the alternating current, allowing a continuous flow of current as the electric field in the air gap displaces and builds up charges on the plates, generating an alternating current. Even though the air acts as an insulator (no physical path) between the plates, it acts as a dielectric medium allowing the crossed electric field to establish a potential difference across the plate. This causes an alternating current to flow through the circuit even in the presence of an air gap. In conclusion, since the air gap acts as a dielectric medium and allows the electric field to establish a potential difference between the plates of the capacitor, it does not open the circuit or act like an open switch. This variation in potential difference and electric field enables a continuous flow of alternating current through the RLC circuit, in spite of the air gap.