A radio capacitor of variable capacitance is made of `n` parallel plates each of area `A` and separated from each other by a distanced. The alternate plates are connected together. The capacitance of the combination is.

Options

A. `(n A in_(o))/(d)`
B. `((n-1) A in_(o))/(d)`
C. `((2n -1)A in_(o))/(d)`
D. `((n-2)A in_(o))/(d)`

Explanation

To find the capacitance of a variable capacitor made of `n` parallel plates, where alternate plates are connected together, we can follow these steps:

### Step 1: Understanding the Configuration
In this setup, we have `n` parallel plates, and alternate plates are connected together. This means that if we have `n` plates, they can be grouped into pairs of connected plates.

### Step 2: Counting the Number of Capacitors
Since alternate plates are connected, we can think of the configuration as consisting of `n/2` capacitors in parallel (if `n` is even). Each capacitor is formed by two adjacent plates.

### Step 3: Capacitance of Each Capacitor
The capacitance \( C \) of a single capacitor formed by two adjacent plates can be calculated using the formula for the capacitance of parallel plates:
\[ C = \frac{\varepsilon_0 A}{d} \]
where:
- \( \varepsilon_0 \) is the permittivity of free space,
- \( A \) is the area of each plate,
- \( d \) is the separation between the plates.

### Step 4: Total Capacitance of the Configuration
Since we have \( n/2 \) capacitors in parallel, the total capacitance \( C_{total} \) of the configuration can be calculated as:
\[ C_{total} = \frac{n}{2} C \]
Substituting the value of \( C \) from Step 3:
\[ C_{total} = \frac{n}{2} \left(\frac{\varepsilon_0 A}{d}\right) \]

### Step 5: Final Expression
Thus, the final expression for the total capacitance of the variable capacitor configuration is:
\[ C_{total} = \frac{n \varepsilon_0 A}{2d} \]

### Summary
The capacitance of the combination of `n` parallel plates, where alternate plates are connected together, is given by:
\[ C_{total} = \frac{n \varepsilon_0 A}{2d} \]

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