Suppose an attractive nuclear force acts between two protons whic… - Vidyadip

Question

Suppose an attractive nuclear force acts between two protons which may be written as $\text{F}=\text{Ce}^{-\text{kr}}/\text{r}^2.$

Write down the dimensional formulae and appropriate $SI$ units of $C$ and $k.$
Suppose that $k = 1$ fermi$^{-1}$ and that the repulsive electric force between the protons is just balanced by the attractive nuclear force when the separation is $5$ fermi. Find the value of $C.$

✓

Answer

Expression of electrical force $\text{F}=\text{C}\times\text{e}^{\frac{-\text{kr}}{\text{r}^2}}$
Since $e^{-kr}$ is a pure number.
So, dimensional formulae of $\text{F}=\frac{\text{dimensional formulae of C}}{\text{dimensional formulae of r}^2}$
Or, $\big[\text{MLT}^{-2}\big]\big[\text{L}^2\big]=$ dimensional formulae of $\text{C}=\big[\text{ML}^3\text{T}^{-2}\big]$
Unit of $C =$ unit of force $\times$ unit of $r^2 =$ Newton $\times$ $m^2 =$ Newton$-m^2$
Since $-kr$ is a number hence dimensional formulae of $\text{k}=\frac{1}{\text{dim entional formulae of r}}=\big[\text{L}^{-1}\big]$ Unit of $k = m^{-1}$

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