The plates of a parallel plate capacitor are pulled apart with a velocity `v`. If at any instant their mutual distance of separation is `x`, then magnitude of rate of change of capacitance with respect to time varies as

Options

A. `(1)/(x)`
B. `(1)/(x^(2))`
C. `x^(2)`
D. `x`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship of capacitance
The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{A \epsilon_0}{x} \]
where:
- \( A \) is the area of the plates,
- \( \epsilon_0 \) is the permittivity of free space,
- \( x \) is the distance between the plates.

### Step 2: Express the distance \( x \) in terms of time \( t \)
Since the plates are being pulled apart with a constant velocity \( v \), the distance \( x \) at any time \( t \) can be expressed as:
\[ x = vt \]

### Step 3: Substitute \( x \) into the capacitance formula
Substituting \( x \) into the capacitance formula gives:
\[ C = \frac{A \epsilon_0}{vt} \]

### Step 4: Differentiate capacitance with respect to time
To find the rate of change of capacitance with respect to time, we differentiate \( C \) with respect to \( t \):
\[ \frac{dC}{dt} = \frac{d}{dt} \left( \frac{A \epsilon_0}{vt} \right) \]
Using the quotient rule for differentiation:
\[ \frac{dC}{dt} = -\frac{A \epsilon_0}{v} \cdot \frac{1}{t^2} \]

### Step 5: Take the magnitude of the rate of change of capacitance
The magnitude of the rate of change of capacitance is:
\[ \left| \frac{dC}{dt} \right| = \frac{A \epsilon_0}{v t^2} \]

### Step 6: Substitute \( t \) back in terms of \( x \)
Since \( x = vt \), we can express \( t \) as \( t = \frac{x}{v} \). Substituting this back gives:
\[ \left| \frac{dC}{dt} \right| = \frac{A \epsilon_0 v}{x^2} \]

### Step 7: Conclude the relationship
From the above expression, we can conclude that the magnitude of the rate of change of capacitance with respect to time varies as:
\[ \left| \frac{dC}{dt} \right| \propto \frac{1}{x^2} \]

### Final Answer
Thus, the magnitude of the rate of change of capacitance with respect to time varies as \( \frac{1}{x^2} \).

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