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

Answer: (B)

$L^2 M^{-1} T^{-1}$

Answer 2

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:

$L^2M^{-1}T.$