Chapter 12: Q19E (page 556)

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

Expert verified

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,

$F_{x} = - \frac{d U}{d x}$

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

$d U = - F_{x} d x$

Calculation of the potential energy

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

$U (x) = - \int F_{x} d x$

For force,the formula can be written as:

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

The energy of this force would be given as,

$U = - \int F d r$

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

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

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 $b > 1$

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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

Step by step solution

Given data

Concept of Potential energy

Calculation of the potential energy

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

Short Answer

Step by step solution

Given data

Concept of Potential energy

Calculation of the potential energy

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Measurements

Fluids

Turning Points in Physics

Wave Optics

Radiation

Study anywhere. Anytime. Across all devices.

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?