Question

The dimensional formula for RC is:

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

Answer 1

The Correct Option is B

To determine the dimensional formula for \(RC\), where \(R\) is resistance and \(C\) is capacitance, we start by identifying their individual dimensional formulas. The resistance \(R\) is given by the formula \(R = V/I\), where \(V\) is voltage and \(I\) is current. The dimensional formula for voltage \(V\) is \([ML^2T^{-3}A^{-1}]\) and for current \(I\) is \([A]\). Therefore, the dimensional formula for resistance \(R\) is:

\[ [R] = [ML^2T^{-3}A^{-1}]/[A] = [ML^2T^{-3}A^{-2}] \]

Capacitance \(C\) is given by the formula \(C = Q/V\), where \(Q\) is charge and \(V\) is voltage. The dimensional formula for charge \(Q\) is \([AT]\), so the dimensional formula for capacitance \(C\) is:

\[ [C] = [AT]/[ML^2T^{-3}A^{-1}] = [M^{-1}L^{-2}T^4A^2] \]

The product \(RC\) is computed by multiplying the dimensional formulas of \(R\) and \(C\):

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

Simplifying, we get:

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

Thus, the correct dimensional formula for \(RC\) simplifies to \([M^0L^0T^1A^0]\).

Answer 2

The time constant $RC$ is the product of resistance $R$ and capacitance $C$.

The dimensional formula for resistance $R$ is $[ML^2T^{-3}A^{-2}]$ and for capacitance $C$, it is $[M^{-1}L^{-2}T^4A^2]$.

Thus, the dimensional formula for $RC$ is:

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

So, the correct answer is $[M^0L^0T^1A^0]$.