Question

The dimensional formula for \( RC \) is:

• A. \( [M L^2 T^{-3} A^{-2}] \)
• B. \( [M^0 L^0 T^{1} A^0] \)
• C. \( [M^1 L^{-2} T^4 A^2] \)
• D. \( [M^0 L^0 T^0 A^1] \)

Hint:

The time constant \( RC \) has the dimensional formula corresponding to time, as it represents the time it takes for a capacitor to charge or discharge through a resistor.

Answer 1

The Correct Option is B

The dimensional formula represents the relationship between different physical quantities. To determine the dimensional formula for \( RC \), where \( R \) stands for resistance and \( C \) for capacitance, we first find the dimensions of \( R \) and \( C \) separately. 

Resistance \((R)\): The dimensional formula for resistance can be derived from Ohm's Law \( V = IR \), where \( V \) (Voltage) has a dimensional formula \([M^1L^2T^{-3}A^{-1}]\) and \( I \) (Current) has a dimensional formula \([A^1]\). Solving for \( R \), it follows that:

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

Capacitance \((C)\): Capacitance is defined by the relation \( Q = CV \), where \( Q \) (Charge) has a dimensional formula \([A^1T^1]\), and rearranging gives us:

\(C = \frac{Q}{V} \Rightarrow [A^1T^1][M^{-1}L^{-2}T^{3}A^{1}] = [M^{-1}L^{-2}T^{4}A^{2}]\)

Given \( RC \) is the product of resistance and capacitance, we multiply their dimensional formulas:

\[RC = [M^1L^2T^{-3}A^{-2}][M^{-1}L^{-2}T^{4}A^{2}] = [M^{1-1}L^{2-2}T^{-3+4}A^{-2+2}] = [M^0L^0T^1A^0]\]

Thus, correct consideration aligns with answer: \([M^0L^0T^{1}A^0]\).