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 × 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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