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Q19E

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Found in: Page 556

### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

# 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 particles Infinitely far is not to be infinite?

The limit on b is $b>1$ .

See the step by step solution

## Step 1: Given data

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

## Step 2: 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,

${d}{U}{=}{-}{{F}}_{{x}}{d}{x}$

## Step 3: 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 $b>1$