Question

In order to measure physical quantities in the sub-atomic world, the quantum theory often employs energy $E$, angular momentum $J$ and velocity $c$ as fundamental dimensions instead of the usual mass, length and time. Then, the dimension of force in this theory is

• A. $\frac{[E{]}^2}{\left[J\right]\left[c\right]}$
• B. $\frac{[E{]}^2}{\left[J{]}^2\right[c]}$
• C. $\frac{\left[E\right]}{\left[J\right]\left[c\right]}$
• D. $\frac{\left[E\right]\left[J\right]}{\left[c\right]}$

Answer 1

The correct option is A $\frac{[E{]}^2}{\left[J\right]\left[c\right]}$

As per the question let the relation be
force $F=k{E}^a{J}^b{c}^c$, where $k$ is any constant.
$\left[ML{T}^{-2}\right]=\left[M{L}^2{T}^{-2}{]}^a\right[M{L}^2{T}^{-1}{]}^b[L{T}^{-1}{]}^c$
$\Rightarrow \left[ML{T}^{-2}\right]=\left[M^{(a+b)}{L}^{(2a+2b+c)}{T}^{(-2a-b-c)}\right]$
On solving the equation by comparing the powers we get
$a=2,b=-1,c=-1$
Thus, the relation will be $\frac{[E{]}^2}{\left[J\right]\left[c\right]}$