If `X` is capacitance and `Y` is the magnetic field which are related by `X=2aY^(2)`. Dimension of a will be:-

Options

A. `[M^(-3)L^(-2)]T^(8)A^(4)]`
B. `[M^(-1)L^(-2)T^(4)A^(2)]`
C. `[M^(-1)L^(0)T^(-4)A^(-2)]`
D. `[M^(-2)L^(-2)T^(6)A^(3)]`

Answer

To find the dimension of \( a \) in the equation \( X = 2aY^2 \), where \( X \) is capacitance and \( Y \) is the magnetic field, we can follow these steps:

### Step 1: Identify the dimensions of \( X \) and \( Y \)

1. **Capacitance \( X \)**: The capacitance \( C \) is defined as the charge per unit voltage:
\[ C = \frac{Q}{V} \]
The dimension of charge \( Q \) is \( [I][T] \) (where \( I \) is current and \( T \) is time) and the dimension of voltage \( V \) is given by:
\[ V = \frac{W}{Q} = \frac{ML^2T^{-2}}{Q} = \frac{ML^2T^{-2}}{IT} \]
Therefore, the dimension of voltage \( V \) is:
\[ [V] = \frac{ML^2T^{-2}}{IT} = [M][L^2][T^{-3}][I^{-1}] \]
Thus, the dimension of capacitance \( X \) is:
\[ [X] = \frac{[Q]}{[V]} = \frac{[I][T]}{[M][L^2][T^{-3}][I^{-1}]} = [M^{-1}][L^{-2}][T^4][I^2] \]

2. **Magnetic Field \( Y \)**: The magnetic field \( B \) can be defined using the Lorentz force:
\[ F = QVB \implies B = \frac{F}{QV} \]
The dimension of force \( F \) is:
\[ [F] = [M][L][T^{-2}] \]
Therefore, the dimension of magnetic field \( Y \) is:
\[ [Y] = \frac{[F]}{[Q][V]} = \frac{[M][L][T^{-2}]}{[I][T][M][L^2][T^{-3}][I^{-1}]} = [M^{0}][L^{-1}][T^{2}][I^{-1}] \]

### Step 2: Rearranging the equation to find \( a \)

From the equation \( X = 2aY^2 \), we can express \( a \) as:
\[ a = \frac{X}{2Y^2} \]

### Step 3: Substitute the dimensions into the equation

Substituting the dimensions we found:
\[ [a] = \frac{[X]}{[Y]^2} = \frac{[M^{-1}][L^{-2}][T^4][I^2]}{([M^{0}][L^{-1}][T^{2}][I^{-1}])^2} \]

Calculating \( [Y]^2 \):
\[ [Y]^2 = [M^{0}][L^{-2}][T^{4}][I^{-2}] \]

### Step 4: Simplifying the dimensions

Now substituting back:
\[ [a] = \frac{[M^{-1}][L^{-2}][T^4][I^2]}{[M^{0}][L^{-2}][T^{4}][I^{-2}]} = [M^{-1}][L^{0}][T^{0}][I^{4}] = [M^{-1}][I^{4}] \]

### Final Answer

Thus, the dimension of \( a \) is:
\[ [a] = [M^{-1}][I^{4}] \]