If L is the inductance, C is the capacitance and R is the resistance, then `Rsqrt(C/L)`. has the dimension

Options

A. `MLT^2 I^(-2)`
B. `ML^2 T^2I`
C. `ML^(-1)T^(-2)I^(-1)`
D. `M^0L^0T^0 I^0`

Answer

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

### Step 1: Write down the dimensions of \( R \), \( C \), and \( L \)

- The dimension of resistance \( R \) is given by:
\[ [R] = M^1 L^2 T^{-3} I^{-2} \]

- The dimension of capacitance \( C \) is given by:
\[ [C] = M^{-1} L^{-2} T^{4} I^{2} \]

- The dimension of inductance \( L \) is given by:
\[ [L] = M^1 L^2 T^{-2} I^{-2} \]

### Step 2: Substitute the dimensions into the expression

We need to find the dimensions of \( R \sqrt{\frac{C}{L}} \):
\[ R \sqrt{\frac{C}{L}} = R \cdot \sqrt{\frac{[C]}{[L]}} \]

### Step 3: Calculate \( \frac{C}{L} \)

Now we will find the dimensions of \( \frac{C}{L} \):
\[ \frac{C}{L} = \frac{M^{-1} L^{-2} T^{4} I^{2}}{M^{1} L^{2} T^{-2} I^{-2}} = M^{-1-1} L^{-2-2} T^{4-(-2)} I^{2-(-2)} = M^{-2} L^{-4} T^{6} I^{4} \]

### Step 4: Take the square root of \( \frac{C}{L} \)

Now, we take the square root:
\[ \sqrt{\frac{C}{L}} = \sqrt{M^{-2} L^{-4} T^{6} I^{4}} = M^{-1} L^{-2} T^{3} I^{2} \]

### Step 5: Multiply by \( R \)

Now we multiply \( R \) with \( \sqrt{\frac{C}{L}} \):
\[ R \sqrt{\frac{C}{L}} = (M^1 L^2 T^{-3} I^{-2}) \cdot (M^{-1} L^{-2} T^{3} I^{2}) \]

### Step 6: Combine the dimensions

Now we combine the dimensions:
\[ = M^{1-1} L^{2-2} T^{-3+3} I^{-2+2} = M^{0} L^{0} T^{0} I^{0} \]

### Final Result

Thus, the dimensions of \( R \sqrt{\frac{C}{L}} \) are:
\[ M^{0} L^{0} T^{0} I^{0} \]

This means the expression is dimensionless, which corresponds to option 4: \( M^0 L^0 T^0 I^0 \).