Which of the following represents the dimensions of Farad

Options

A. `M^(-1)L^(-2)T^(4)A^(2)`
B. `ML^(2)T^(2)A^(-2)`
C. `ML^(2)T^(2)A^(-1)`
D. `MT^(-2)A^(-1)`

Answer

To find the dimensions of the unit Farad (F), which is the unit of capacitance, we can follow these steps:

### Step 1: Understand the definition of capacitance
Capacitance (C) is defined as the amount of charge (Q) stored per unit voltage (V). Therefore, we can express capacitance as:
\[ C = \frac{Q}{V} \]

### Step 2: Express voltage in terms of work done and charge
Voltage (V) can be defined as the work done (W) per unit charge (Q):
\[ V = \frac{W}{Q} \]
Thus, we can rearrange this to express work done:
\[ W = Q \cdot V \]

### Step 3: Substitute the expression for voltage into the capacitance formula
Substituting the expression for voltage into the capacitance formula gives us:
\[ C = \frac{Q}{\frac{W}{Q}} = \frac{Q^2}{W} \]

### Step 4: Determine the dimensions of charge and work done
- Charge (Q) has the dimension of current (I) multiplied by time (T):
\[ [Q] = [I][T] = [A][T] \]
- Work done (W) has the dimension of energy, which can be expressed as force multiplied by distance. The dimension of force is mass (M) times acceleration (L/T²):
\[ [W] = [F] \cdot [d] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]

### Step 5: Substitute the dimensions into the capacitance formula
Now substituting the dimensions back into the capacitance formula:
\[ [C] = \frac{[Q]^2}{[W]} = \frac{([A][T])^2}{[M][L^2][T^{-2}]} \]
\[ = \frac{[A^2][T^2]}{[M][L^2][T^{-2}]} \]

### Step 6: Simplify the expression
Now, simplifying this gives:
\[ [C] = \frac{[A^2][T^2][T^2]}{[M][L^2]} = \frac{[A^2][T^4]}{[M][L^2]} \]

### Final Result
Thus, the dimensions of Farad (F) are:
\[ [F] = \frac{[A^2][T^4]}{[M][L^2]} \]