Question

Consider two physical quantities \( A \) and \( B \) related to each other as \( E = \frac{B - x^2}{At} \) where \( E \), \( x \), and \( t \) have dimensions of energy, length, and time, respectively. The dimension of \( AB \) is:

• A. \( L^{-2} M T^0 \)
• B. \( L^2 M^{-1} T^{-1} \)
• C. \( L^{-2} M^{-1} T^1 \)
• D. \( L^0 M^{-1} T^1 \)

Answer 1

The Correct Option is B

Given:

\[ E = \frac{B - x^2}{At}. \]

The dimensions of \(E\), \(x\), and \(t\) are:

\[ [E] = ML^2T^{-2}, \quad [x] = L, \quad [t] = T. \]

The term \(B - x^2\) must have the same dimensions as \(E\), so:

\[ [B] = L^2. \]

Rearrange the equation to find the dimensions of \(A\):

\[ A = \frac{B - x^2}{E \cdot t} = \frac{L^2}{ML^2T^{-2} \cdot T} = M^{-1}T. \]

Therefore:

\[ [A] = M^{-1}T. \]

The dimensions of \(AB\) are:

\[ [AB] = [A][B] = (M^{-1}T)(L^2) = L^2M^{-1}T. \]

Thus, the answer is:

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