If R is the electric resistance and C is the capacitance, the dimensional formula of RC is

Options

A. `LT^(-1)`
B. `T^(-1)`
C. `MLT^(-2)`
D. `T`

Answer

To find the dimensional formula of the product \( RC \) where \( R \) is the electric resistance and \( C \) is the capacitance, we will follow these steps:

### Step 1: Write the formulas for resistance \( R \) and capacitance \( C \).

- The formula for electric resistance \( R \) is given by:
\[ R = \frac{V}{I} \]
where \( V \) is the potential difference (voltage) and \( I \) is the electric current.

- The formula for capacitance \( C \) is given by:
\[ C = \frac{Q}{V} \]
where \( Q \) is the electric charge.

### Step 2: Write the dimensional formulas for \( R \) and \( C \).

- The dimensional formula for potential difference \( V \) can be expressed as:
\[ V = \frac{W}{Q} = \frac{ML^2T^{-2}}{T \cdot A} = \frac{ML^2T^{-2}}{T \cdot A} = ML^2T^{-3}A^{-1} \]

- Therefore, the dimensional formula for resistance \( R \) becomes:
\[ R = \frac{V}{I} = \frac{ML^2T^{-3}A^{-1}}{A} = ML^2T^{-3}A^{-2} \]

- The dimensional formula for charge \( Q \) is:
\[ Q = IT = A \cdot T \]

- Thus, the dimensional formula for capacitance \( C \) is:
\[ C = \frac{Q}{V} = \frac{AT}{ML^2T^{-3}A^{-1}} = \frac{AT}{ML^2T^{-3}A^{-1}} = \frac{AT^4}{ML^2} \]

### Step 3: Combine the dimensional formulas of \( R \) and \( C \).

Now we can find the dimensional formula of the product \( RC \):
\[ RC = R \cdot C = (ML^2T^{-3}A^{-2}) \cdot \left(\frac{AT^4}{ML^2}\right) \]

### Step 4: Simplify the expression.

When we multiply the two dimensional formulas:
\[ RC = ML^2T^{-3}A^{-2} \cdot \frac{AT^4}{ML^2} \]

- The \( M \) cancels out:
\[ = L^2T^{-3}A^{-2} \cdot \frac{AT^4}{L^2} \]
- The \( L^2 \) cancels out:
\[ = T^{-3}A^{-2} \cdot AT^4 \]
- The \( A \) cancels out:
\[ = T^{-3}T^4A^{-1} = T^{1}A^{-1} \]

### Final Result

Thus, the dimensional formula for \( RC \) is:
\[ T^1A^{-1} \]