Question

Suppose a force between two particles decreases distance according to $F=k/r^b$. What is the limit on b if the energy required to separate the particlesInfinitely far is not to be infinite?

Short answer

The limit on b is $b>1$.

Step-by-step solution

Given data

A force between two particles decreases with distance, according to $F=k/r^b$.

Concept of Potential energy

The force $F_x$ is the negative of the derivative of the potential energy U such that,

$F_x=-\frac{dU}{dx}$

Therefore, the differential form of the potential energy can be written as,

$dU=-F_{x}dx$

Calculation of the potential energy

Integrate the above equation on both sides, andgetthe expression for the total potential energy as:

$U\left(x\right)=-\int F_{x}dx$

For force,the formula can be written as:

$F=\frac{k}{r^b}$

The energy of this force would be given as,

$U=-\int Fdr$

Substitute $F=\frac{k}{r^b}$in the above equation as:

$\begin{array}{rcl}U& =& \int \frac{k}{r^b}dr\\ U& =& -\int k{r}^{-b}dr\\ U& =& -\frac{k{r}^{-b+1}}{-b+1}\\ & & \end{array}$

When b is not 1 , we need $-b+1<0$to make sure the energy is not infinite at infinite separation, which is $b>1$. Thus, the limit on is $b>1$