Find the dimensions of capacitance.

Options

A. `[M^(-1) L^(-2) T A^(2)]`
B. `[M^(-1) L^(-2) T^(3) A^(2)]`
C. `[M^(-1) L^(-2) T^(4) A^(2)]`
D. `[M^(-1) L^(-2) T^(2) A^(2)]`

Explanation

To find the dimensions of capacitance, we will follow these steps:

### Step 1: Understand the relationship between charge, capacitance, and voltage
The formula for capacitance (C) is given by the equation:
\[ C = \frac{Q}{V} \]
where \( Q \) is the charge stored in the capacitor and \( V \) is the voltage across the capacitor.

### Step 2: Find the dimensions of charge (Q)
We know that current (I) is defined as the rate of flow of charge:
\[ I = \frac{Q}{t} \]
From this, we can express charge as:
\[ Q = I \cdot t \]
The dimension of current (I) is denoted as \( [I] = A \) (where A is amperes), and the dimension of time (t) is denoted as \( [t] = T \). Therefore, the dimension of charge (Q) is:
\[ [Q] = [I] \cdot [t] = A \cdot T = AT \]

### Step 3: Find the dimensions of voltage (V)
Voltage (V) is defined as the work done (W) per unit charge (Q):
\[ V = \frac{W}{Q} \]
To find the dimension of voltage, we first need the dimension of work done. Work done is defined as force (F) multiplied by displacement (d):
\[ W = F \cdot d \]
The dimension of force (F) is given by:
\[ [F] = [M][L][T^{-2}] \]
Thus, the dimension of work done is:
\[ [W] = [F] \cdot [d] = [M][L][T^{-2}] \cdot [L] = [M][L^2][T^{-2}] \]

Now substituting the dimensions of work done and charge into the voltage equation:
\[ [V] = \frac{[W]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[A][T]} \]
This simplifies to:
\[ [V] = [M][L^2][T^{-3}][A^{-1}] \]

### Step 4: Substitute dimensions of Q and V into the capacitance formula
Now we can substitute the dimensions of charge and voltage back into the capacitance formula:
\[ [C] = \frac{[Q]}{[V]} = \frac{[AT]}{[M][L^2][T^{-3}][A^{-1}]} \]
This simplifies to:
\[ [C] = [AT] \cdot \frac{[A]}{[M][L^2][T^{-3}]} = \frac{[A^2][T]}{[M][L^2][T^{-3}]} \]
Combining the terms gives us:
\[ [C] = \frac{[A^2][T^4]}{[M][L^2]} \]

### Step 5: Final dimensions of capacitance
Thus, the final dimensions of capacitance (C) are:
\[ [C] = [M^{-1}][L^{-2}][T^4][A^2] \]

### Summary of the dimensions of capacitance:
\[ [C] = M^{-1} L^{-2} T^4 A^2 \]