Inductance L can be dimensional represented as

Options

A. `ML^(2)T^(-2)A^(-1)`
B. `ML^(2)T^(-4)A^(-3)`
C. `ML^(-2)T^(-2)A^(-2)`
D. `ML^(2)T^(4)A^(3)`

Answer

To find the dimensional representation of inductance \( L \), we can follow these steps:

### Step 1: Understand the relationship involving inductance
Inductance \( L \) is related to the inductive reactance \( X_L \), which can be expressed as:
\[ X_L = \frac{V}{I} \]
where \( V \) is voltage and \( I \) is current.

### Step 2: Write the dimensional formula for voltage \( V \)
Voltage \( V \) can be expressed in terms of work \( W \) and charge \( Q \):
\[ V = \frac{W}{Q} \]
The dimensional formula for work \( W \) is:
\[ W = ML^2T^{-2} \]
And the dimensional formula for charge \( Q \) is:
\[ Q = IT = AT \]
where \( A \) is the unit of current.

### Step 3: Substitute the dimensional formulas into the voltage equation
Now substituting these into the voltage equation:
\[ V = \frac{ML^2T^{-2}}{AT} \]
This gives us:
\[ V = ML^2T^{-3}A^{-1} \]

### Step 4: Write the dimensional formula for current \( I \)
The dimensional formula for current \( I \) is simply:
\[ I = A \]

### Step 5: Find the dimensional formula for inductive reactance \( X_L \)
Now substituting the dimensional formulas for \( V \) and \( I \) into the equation for \( X_L \):
\[ X_L = \frac{V}{I} = \frac{ML^2T^{-3}A^{-1}}{A} = ML^2T^{-3}A^{-2} \]

### Step 6: Relate inductive reactance \( X_L \) to inductance \( L \)
The inductive reactance \( X_L \) is also related to inductance \( L \) and angular frequency \( \omega \):
\[ X_L = \omega L \]
where \( \omega \) has the dimensional formula:
\[ \omega = \frac{2\pi}{T} \quad \text{(dimensionally, it is } T^{-1}\text{)} \]

### Step 7: Rearranging to find the dimensional formula for \( L \)
From the equation \( X_L = \omega L \), we can express \( L \) as:
\[ L = \frac{X_L}{\omega} \]
Substituting the dimensional formulas:
\[ L = \frac{ML^2T^{-3}A^{-2}}{T^{-1}} = ML^2T^{-2}A^{-2} \]

### Conclusion
Thus, the dimensional representation of inductance \( L \) is:
\[ [L] = ML^2T^{-2}A^{-2} \]

### Final Answer
The correct option is \( ML^2T^{-2}A^{-2} \).

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