Question

What is the dimension of \[ \frac{M B}{K T} \] where \( M \) represents magnetic moment, \( K \) represents the Boltzmann constant, \( B \) represents the magnetic field, and \( T \) represents temperature?

• A. \( \text{[M]} \cdot \text{[T]}^2 \)
• B. \( \text{[M]} \cdot \text{[K]} \)
• C. \( \text{[M]} \cdot \text{[K]}^{-1} \)
• D. \( \text{[M]} \cdot \text{[T]}^{-1} \)

Hint:

When dealing with dimensions, make sure to break down each quantity into its fundamental dimensions (Mass, Length, Time, etc.) and use dimensional analysis to simplify the expression.

Answer 1

The Correct Option is C

The dimensions of each quantity are as follows: - \( M \) (magnetic moment) has the dimension of \( \text{[M]} \cdot \text{[L]}^2 \cdot \text{[T]}^{-1} \) - \( B \) (magnetic field) has the dimension of \( \text{[M]} \cdot \text{[T]}^{-2} \cdot \text{[I]}^{-1} \) - \( K \) (Boltzmann constant) has the dimension of \( \text{[M]} \cdot \text{[L]}^2 \cdot \text{[T]}^{-2} \cdot \text{[K]}^{-1} \) - \( T \) (temperature) has the dimension of \( \text{[K]} \) Thus, the overall dimension of \( \frac{M B}{K T} \) is: \[ \frac{\text{[M]} \cdot \text{[L]}^2 \cdot \text{[T]}^{-1} \cdot \text{[M]} \cdot \text{[T]}^{-2} \cdot \text{[I]}^{-1}}{\text{[M]} \cdot \text{[L]}^2 \cdot \text{[T]}^{-2} \cdot \text{[K]}^{-1} \cdot \text{[K]}} \] Simplifying, we get: \[ \text{[M]} \cdot \text{[K]}^{-1} \]