The variable capacitance of an old radio tuner consists of four plates connected together placed alternately between four other plates, also connected together (Fig. 24–36). Each plate is separated from its neighbor by 1.6 mm of air. One set of plates can move so that the area of overlap of each plate varies from 2.0 cm² to 9.0 cm².
(a) Are these seven capacitors connected in series or in parallel?
(b) Determine the range of capacitance values.

Verified step by step guidance

1. Step 1: Analyze the configuration of the plates. The image shows alternating plates connected together, forming a system where each pair of adjacent plates acts as a capacitor. Since all capacitors share the same potential difference (voltage), they are connected in parallel.
2. Step 2: Recall the formula for the capacitance of a parallel plate capacitor: $C=\frac{\epsilon }{d}\cdot A$, where $\epsilon$ is the permittivity of the medium (air in this case), $d$ is the separation between the plates, and $A$ is the area of overlap between the plates.
3. Step 3: Substitute the given values into the formula. The separation between the plates is $1.6$ mm ($1.6\; \times \; 10-3$ m), and the permittivity of air is approximately $8.85\; \times \; 10-12$ F/m. The area of overlap varies from $2.0$ cm² ($2.0\; \times \; 10-4$ m²) to $9.0$ cm² ($9.0\; \times \; 10-4$ m²).
4. Step 4: Calculate the capacitance for the minimum overlap area using the formula. Use $A\; =\; 2.0\; \times \; 10-4$ m² and substitute into the formula to find the minimum capacitance.
5. Step 5: Calculate the capacitance for the maximum overlap area using the formula. Use $A\; =\; 9.0\; \times \; 10-4$ m² and substitute into the formula to find the maximum capacitance. This gives the range of capacitance values.

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is defined by the formula C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the plates. The unit of capacitance is the farad (F), and it depends on the physical characteristics of the capacitor, including the area of the plates and the distance between them.

Series and Parallel Connections

In electrical circuits, capacitors can be connected in series or parallel. In a series connection, the total capacitance decreases and is given by the reciprocal of the sum of the reciprocals of individual capacitances. In contrast, in a parallel connection, the total capacitance is the sum of the individual capacitances, leading to an increase in total capacitance. Understanding these configurations is crucial for analyzing how capacitors behave in a circuit.

Variable Capacitance

Variable capacitance refers to capacitors whose capacitance can be adjusted, often through mechanical means, such as moving plates. In the context of the radio tuner described, the overlapping area of the plates changes, which alters the capacitance. This property is essential for tuning frequencies in radios, as it allows for the selection of different stations by varying the capacitance.