Physics - Unit & Dimension Question with Solution | TestHub

To find the dimensional formula of $\beta$, we use the given relation and the dimensions of each variable.

Given relation: $\beta =\frac{k\theta }{PZ}$

Dimensions of the variables:

Boltzmann constant ($k$): $[M{L}^2{T}^{-2}{\Theta }^{-1}]$ (from $E=kT$, where $E$ is energy)

Temperature ($\theta$): $[\Theta ]$

Pressure ($P$): $[M{L}^{-1}{T}^{-2}]$ (from $P=\frac{F}{A}=\frac{ML{T}^{-2}}{L^2}$)

Distance ($Z$): $[L]$

Substitute these dimensions into the relation for $\beta$:

$$\beta =\frac{[M{L}^2{T}^{-2}{\Theta }^{-1}][\Theta ]}{[M{L}^{-1}{T}^{-2}][L]}$$

$$\beta =\frac{[M{L}^2{T}^{-2}]}{[M{L}^0{T}^{-2}]}$$

$$\beta =[M^{1-1}{L}^{2-0}{T}^{-2-(-2)}]$$

$$\beta =[M^0{L}^2{T}^0]$$

The dimensional formula of $\beta$ is $[M^0{L}^2{T}^0]$.

The final answer is $\boxed{[M^0{L}^2{T}^0]}$.