Which of the following pairs has the same dimensions?

Options

A. Torque and work
B. Angular momentum and Planck's constant
C. Energy momentum and Planck's constant
D. Light year and wavelength

Explanation

To determine which pairs of physical quantities have the same dimensions, we will analyze each pair step by step.

### Step 1: Analyze Torque and Work
- **Torque (τ)**: Torque is defined as the cross product of force (F) and distance (r).
\[ τ = F \times r \]
- The dimension of force (F) is given by:
\[ [F] = MLT^{-2} \]
- The dimension of distance (r) is:
\[ [r] = L \]
- Therefore, the dimension of torque is:
\[ [τ] = [F] \times [r] = (MLT^{-2}) \times (L) = ML^2T^{-2} \]

- **Work (W)**: Work is defined as the dot product of force and displacement (d).
\[ W = F \cdot d \]
- The dimension of displacement (d) is also:
\[ [d] = L \]
- Therefore, the dimension of work is:
\[ [W] = [F] \times [d] = (MLT^{-2}) \times (L) = ML^2T^{-2} \]

Since both torque and work have the same dimensions \(ML^2T^{-2}\), they are dimensionally equivalent.

### Step 2: Analyze Angular Momentum and Planck's Constant
- **Angular Momentum (L)**: Angular momentum is defined as:
\[ L = mvr \]
- Here, mass (m) has dimensions:
\[ [m] = M \]
- Velocity (v) has dimensions:
\[ [v] = LT^{-1} \]
- Therefore, the dimension of angular momentum is:
\[ [L] = [m] \times [v] \times [r] = M \times (LT^{-1}) \times (L) = ML^2T^{-1} \]

- **Planck's Constant (h)**: Planck's constant is defined as:
\[ h = \lambda \cdot p \]
- Wavelength (λ) has dimensions:
\[ [λ] = L \]
- Momentum (p) has dimensions:
\[ [p] = MLT^{-1} \]
- Therefore, the dimension of Planck's constant is:
\[ [h] = [λ] \times [p] = (L) \times (MLT^{-1}) = ML^2T^{-1} \]

Since both angular momentum and Planck's constant have the same dimensions \(ML^2T^{-1}\), they are dimensionally equivalent.

### Step 3: Analyze Energy and Young's Modulus
- **Energy (E)**: The dimension of energy is the same as work:
\[ [E] = ML^2T^{-2} \]

- **Young's Modulus (Y)**: Young's modulus is defined as stress divided by strain.
- Stress is defined as force per unit area:
\[ [\text{Stress}] = \frac{[F]}{[\text{Area}]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]
- Strain is dimensionless (it is a ratio of lengths).
- Therefore, the dimension of Young's modulus is:
\[ [Y] = [\text{Stress}] = ML^{-1}T^{-2} \]

Since energy has dimensions \(ML^2T^{-2}\) and Young's modulus has dimensions \(ML^{-1}T^{-2}\), they are not dimensionally equivalent.

### Step 4: Analyze Light Year and Wavelength
- **Light Year**: A light year is a measure of distance, specifically the distance light travels in one year. Therefore, its dimension is:
\[ [\text{Light Year}] = L \]

- **Wavelength (λ)**: The dimension of wavelength is also:
\[ [λ] = L \]

Since both light year and wavelength have the same dimensions \(L\), they are dimensionally equivalent.

### Conclusion
The pairs that have the same dimensions are:
- Torque and Work
- Angular Momentum and Planck's Constant
- Light Year and Wavelength

However, Energy and Young's Modulus do not have the same dimensions.