Question

If $ C $ be the capacitance and $ V $ be the electric potential, then the dimensional formula of $ CV^2 $ is

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

Hint:

For dimensional analysis, remember to use the dimensional formulas for basic quantities like charge, potential, and capacitance. Multiply or divide these to get the dimensional formula for more complex expressions.

Answer 1

The Correct Option is A

We know the dimensional formulas of capacitance \( C \) and electric potential \( V \):
1. Capacitance \( C \): The dimensional formula for capacitance is: \[ C = \frac{Q}{V} \] where \( Q \) is charge and \( V \) is electric potential. The dimensional formula for charge \( Q \) is \( [A T] \), and the dimensional formula for electric potential \( V \) is \( [ML^2 T^{-3} A^{-1}] \).
Thus, the dimensional formula for capacitance \( C \) is: \[ [C] = \frac{[A T]}{[ML^2 T^{-3} A^{-1}]} = [M^{-1} L^{-2} T^4 A^2] \]
2. Electric potential \( V \): The dimensional formula for electric potential is already provided as: \[ [V] = [ML^2 T^{-3} A^{-1}] \] Now, the expression \( CV^2 \) has dimensions: \[ [C V^2] = [M^{-1} L^{-2} T^4 A^2] \times [ML^2 T^{-3} A^{-1}]^2 \] Simplifying: \[ [C V^2] = [M^{-1} L^{-2} T^4 A^2] \times [M^2 L^4 T^{-6} A^{-2}] \] \[ [C V^2] = [M^{1} L^2 T^{-2} A^0] \]
Thus, the dimensional formula of \( CV^2 \) is: \[ \text{(A) } [MLT^{-2}A^0] \] Therefore, the correct answer is: \[ \text{(A) } [MLT^{-2}A^0] \]

Answer 2

To determine the dimensional formula of the expression \( CV^2 \), we need to understand the dimensions of each component involved:

• Capacitance (\( C \)): The SI unit of capacitance is the farad (F), which is defined as coulombs per volt. The dimensional formula for a farad can be derived as follows:Capacitance, \( C = \frac{Q}{V} \) where \( Q \) is electric charge (\( [IT] \)) and \( V \) is electric potential (\( [ML^2T^{-3}A^{-1}] \)). Therefore, \( C = \frac{IT}{ML^2T^{-3}A^{-1}} = [M^{-1}L^{-2}T^4A^2] \).
• Electric Potential (\( V \)): Electric potential has the dimensional formula \( [ML^2T^{-3}A^{-1}] \).

Now, consider the expression \( CV^2 \):

\( CV^2 = C \times V \times V \).

Substituting the dimensional formulas, we get:

\( CV^2 = [M^{-1}L^{-2}T^4A^2] \times [ML^2T^{-3}A^{-1}] \times [ML^2T^{-3}A^{-1}] \).

Simplifying the expression by multiplying the dimensions:

\( CV^2 = M^{-1}L^{-2}T^4A^2 \times M^2L^4T^{-6}A^{-2} = M^{1}L^{2}T^{-2}A^{0} \).

Therefore, the dimensional formula for \( CV^2 \) is \( [MLT^{-2}A^0] \).

Option	Dimensional Formula
1	\([MLT^{-2}A^0]\)
2	\([MLT^{-2}A^{-1}]\)
3	\([M^{1}L^{2}T^{-2}A^0]\)
4	\([ML^{-3}T^{1}A]\)

The correct option is 1: \([MLT^{-2}A^{0}]\).