If L and R denote inductance and resistance respectively, then the dimension of `L//R` is :

Options

A. `[M^(0)L^(0)T^(0)]`
B. `[M^(0)L^(0)T]`
C. `[M^(2)L^(0)T^(2)]`
D. `[MLT^(2)]`

Answer

To find the dimension of \( \frac{L}{R} \), where \( L \) is inductance and \( R \) is resistance, we will follow these steps:

### Step 1: Write the dimensions of inductance \( L \).
Inductance \( L \) can be expressed in terms of energy \( U \) and current \( I \). The formula for energy stored in an inductor is given by:

\[ U = \frac{1}{2} L I^2 \]

From this, we can rearrange to find \( L \):

\[ L = \frac{2U}{I^2} \]

Now, we need the dimensions of \( U \) (energy) and \( I \) (current).

The dimension of energy \( U \) is:

\[ [M L^2 T^{-2}] \]

The dimension of current \( I \) is:

\[ [A] \]

Thus, substituting these dimensions into the expression for \( L \):

\[ \text{Dimension of } L = \frac{[M L^2 T^{-2}]}{[A^2]} = [M L^2 T^{-2} A^{-2}] \]

### Step 2: Write the dimensions of resistance \( R \).
Resistance \( R \) is defined as:

\[ R = \frac{V}{I} \]

Where \( V \) (voltage) can be expressed in terms of work done \( W \) and charge \( Q \):

\[ V = \frac{W}{Q} \]

And since work \( W \) is energy, we have:

\[ V = \frac{U}{Q} \]

The dimension of charge \( Q \) is:

\[ [I \cdot T] = [A \cdot T] \]

Thus, the dimension of voltage \( V \) becomes:

\[ \text{Dimension of } V = \frac{[M L^2 T^{-2}]}{[A \cdot T]} = [M L^2 T^{-3} A^{-1}] \]

Now substituting this into the expression for \( R \):

\[ \text{Dimension of } R = \frac{[M L^2 T^{-3} A^{-1}]}{[A]} = [M L^2 T^{-3} A^{-2}] \]

### Step 3: Calculate the dimensions of \( \frac{L}{R} \).
Now, we can find the dimension of \( \frac{L}{R} \):

\[ \frac{L}{R} = \frac{[M L^2 T^{-2} A^{-2}]}{[M L^2 T^{-3} A^{-2}]} \]

When we divide these dimensions, we can cancel out the common terms:

\[ = \frac{[M L^2 T^{-2} A^{-2}]}{[M L^2 T^{-3} A^{-2}]} = [T^{1}] \]

Thus, the dimension of \( \frac{L}{R} \) is:

\[ [M^0 L^0 T^1] = [T] \]

### Final Answer:
The dimension of \( \frac{L}{R} \) is \( [T] \).

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