Question

Which one of the following is the correct dimensional formula for the capacitance in F? M, L, T, and C stand for unit of mass, length, time, and charge.

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

Hint:

To determine the dimensional formula of capacitance, use the relationship \( C = \frac{Q}{V} \), and express the units of \( Q \) and \( V \) in terms of mass, length, time, and charge.

Answer 1

The Correct Option is A

The capacitance \( C \) is defined by the relationship: \[ C = \frac{Q}{V} \] where \( Q \) is the charge and \( V \) is the potential difference. The units for \( Q \) are coulombs \( [C] \), and the units for potential \( V \) are derived from the formula \( V = \frac{U}{Q} = \frac{J}{C} \), where \( J \) (Joules) is energy. The units of energy are \( [M L^2 T^{-2}] \), so: \[ V = \frac{[M L^2 T^{-2}]}{[C]} \] Thus, the dimensional formula for capacitance is: \[ C = \frac{[C]}{[M L^2 T^{-2}] [C]} = [CM^{-1}L^{-2}T^2]. \] Thus, the correct answer is \( \boxed{[CM^{-1}L^{-2}T^2]} \).

Answer 2

Step 1: Start from the definition of capacitance.
Capacitance \( C \) is defined as: \[ C = \frac{Q}{V} \] where \( Q \) = charge and \( V \) = potential difference.

Step 2: Write the dimensional formula of each term.
- Charge \( Q \): \( [Q] = [C] \) (by definition). - Potential difference \( V = \frac{W}{Q} = \frac{\text{Energy}}{\text{Charge}}. \)

Energy (or work) has dimensional formula: \[ [W] = [M L^{2} T^{-2}]. \] Therefore, \[ [V] = \frac{[M L^{2} T^{-2}]}{[C]} = [M L^{2} T^{-2} C^{-1}]. \]

Step 3: Dimensional formula of capacitance.
\[ [C] = \frac{[Q]}{[V]} = \frac{[C]}{[M L^{2} T^{-2} C^{-1}]} = [C^{2} M^{-1} L^{-2} T^{2}]. \]

Step 4: Simplify the expression.
\[ [C] = [C M^{-1} L^{-2} T^{2}] \quad \text{(since C already represents charge unit)}. \]

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Final Answer:

\[ \boxed{[C M^{-1} L^{-2} T^{2}]} \]