Question

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

• A. $\frac{[E{]}^4}{\left[J{]}^3\right[c{]}^3}$
• B. $\frac{[E{]}^2}{\left[J\right]\left[c\right]}$
• C. $\frac{[E{]}^4}{\left[J{]}^2\right[c{]}^2}$
• D. $\frac{[E{]}^3}{\left[J{]}^2\right[c{]}^2}$

Answer 1

The correct option is A $\frac{[E{]}^4}{\left[J{]}^3\right[c{]}^3}$

The unit of pressure is N/m${}^2$ and its dimension in terms of mass, length and time is [M L${}^{-1}$ T${}^{-2}$]

In the above mentioned dimension the entities M, L and T are defined in terms of energy, speed and angular momentum as follows.

Mass = 1 GeVc${}^2$ = 1.782 $\times$ 10${}^{-27}$ kg

Length = 1hc/GeV = 1.973 $\times$ 10${}^{-16}$ m

Time = 1h/GeV = 6.582 $\times$ 10${}^{-26}$ s

In terms of Energy and linear momentum it is defined as,

Energy = 1GeV = 1.602 $\times$ 10${}^{-10}$ J

Linear momentum = 1GeV/c = 5.344 $\times$ 10${}^{-19}$ kg m s${}^{-1}$

The formula of pressure is P = $\frac{F}{A}$ and in terms of G,e and V are i.e energy, speed and angular momentum is (GeV)${}^4$(ch)${}^3$

i.e the entity force is defined as (GeV)${}^4$ and area as (ch)${}^3$.

Hence representing the above dimension in terms of Energy, Angular momentum and velocity we get [E${}^4$ J${}^{-3}$ C${}^{-3}$]