Electric Charge and Electric Field

4 Coulomb’s Law

Learning Objectives

  • State Coulomb's law and describe how the electrostatic force depends on the charges and the distance between them.
  • Calculate the electrostatic force between two charged particles or objects.
  • Compare the strength of the electrostatic force with the gravitational force.

Figure 4.1 reminds us that different fundamental forces dominate at different scales.

Two interacting spiral galaxies illustrating that gravity dominates the motion of very large objects, while electrostatic forces dominate at atomic scales.
Figure 4.1: Gravity shapes the motion of enormous objects such as galaxies. At the atomic scale, however, the electrostatic force between charged particles is vastly stronger than gravity and is responsible for holding atoms and molecules together. (credit: NASA/HST)

Coulomb's Law

Why do two charged objects sometimes attract and sometimes repel? How does the force change as the objects move farther apart? These questions led scientists to search for a quantitative law describing the interaction between electric charges.

By the late eighteenth century, experiments had established several important facts about electric charge. There are two types of charge—positive and negative. Like charges repel, unlike charges attract, and the strength of the electric force decreases as the distance between charges increases. Careful measurements by the French physicist Charles-Augustin de Coulomb (1736–1806) revealed the mathematical relationship that describes this force. Today, this relationship is known as Coulomb's law.

Coulomb's law is one of the fundamental laws of physics. It allows us to predict the electric force between charged particles ranging from electrons and protons to charged objects encountered in everyday life. As we will see, the mathematical form of Coulomb's law closely resembles Newton's law of universal gravitation, although the two forces differ greatly in both strength and behavior.

Electrostatic forces are especially important in the health sciences. They help determine how proteins fold into their functional shapes, how DNA maintains its structure, how ions move through cell membranes, and how many drugs bind to their biological targets. Electrostatic interactions also play an essential role in medical technologies such as electrocardiography (ECG), electrophoresis, and many forms of biomedical sensing and imaging.

The magnitude of the electrostatic force between two point charges is given by

[latex]F=k\frac{|q_1q_2|}{r^2}[/latex]

where:

  • F is the magnitude of the electrostatic force,
  • q1 and q2 are the two point charges,
  • r is the distance between the charges, and
  • k is Coulomb's constant, [latex]k=8.99\times10^9\;\text{N}\cdot\text{m}^2/\text{C}^2[/latex]

Coulomb's law gives the magnitude of the electric force. The force itself is a vector and always acts along the straight line joining the two charges. Like charges repel one another, whereas unlike charges attract. In every interaction, the forces on the two charges are equal in magnitude and opposite in direction, in agreement with Newton's Third Law of Motion. Figure 4.2 summarizes these directions for like and unlike charges.

Although the equation appears simple, establishing it experimentally was a remarkable achievement. Charles-Augustin de Coulomb measured extremely small electric forces using a sensitive instrument called a torsion balance. His experiments showed that the force decreases with the square of the distance between the charges.

This inverse-square dependence can be written as

[latex]F\propto\frac{1}{r^2}[/latex]

This relationship is known as an inverse-square law. The same mathematical dependence appears in Newton's law of gravitation, although the electric force is enormously stronger than gravity at the atomic scale.

Diagram showing that like charges repel and unlike charges attract, with equal and opposite forces acting along the line between the charges.
Figure 4.2: Coulomb's law describes the force between two point charges. Like charges repel, whereas unlike charges attract. The forces acting on the two charges are always equal in magnitude and opposite in direction.

Example 4.1: Comparing the Electric and Gravitational Forces

An electron and a proton in a hydrogen atom are separated by an average distance of [latex]5.30\times10^{-11}\ \text{m}[/latex]. Compare the magnitude of the electrostatic force between them with the magnitude of the gravitational force.

Strategy

We first calculate the electric force using Coulomb's law. We then calculate the gravitational force using Newton's law of universal gravitation. Finally, we compare the two forces by taking their ratio.

Solution

The electrostatic force between the electron and the proton is

[latex]F_e=k\frac{|q_1q_2|}{r^2}[/latex]

Substituting the known values gives

[latex]\begin{aligned} F_e&=(8.99\times10^9) \frac{(1.60\times10^{-19}\,\text{C})(1.60\times10^{-19}\,\text{C})} {(5.30\times10^{-11}\,\text{m})^2}\\[4pt] &=8.19\times10^{-8}\ \text{N} \end{aligned}[/latex]

The gravitational force between the same two particles is

[latex]F_g=G\frac{m_pm_e}{r^2}[/latex]

where [latex]G=6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2[/latex], [latex]m_p=1.67\times10^{-27}\ \text{kg}[/latex], and [latex]m_e=9.11\times10^{-31}\ \text{kg}[/latex]

Substituting these values gives

[latex]\begin{aligned} F_g&=(6.67\times10^{-11}) \frac{(1.67\times10^{-27}\,\text{kg})(9.11\times10^{-31}\,\text{kg})} {(5.30\times10^{-11}\,\text{m})^2}\\[4pt] &=3.63\times10^{-47}\ \text{N} \end{aligned}[/latex]

The ratio of the two forces is therefore

[latex]\begin{aligned} \frac{F_e}{F_g} &=\frac{8.19\times10^{-8}} {3.63\times10^{-47}}\\[4pt] &=2.26\times10^{39} \end{aligned}[/latex]

Discussion

The electrostatic force between an electron and a proton is approximately [latex]2\times10^{39}[/latex] times stronger than their gravitational attraction. This enormous difference explains why gravity is essentially irrelevant at atomic and molecular scales. Instead, electrostatic forces determine the structure of atoms, chemical bonding, protein folding, DNA organization, and the electrical behavior of cells. Gravity dominates only when extremely large masses—such as planets, stars, and galaxies—are involved.

At macroscopic scales, gravity dominates because most objects are electrically neutral overall. The positive and negative charges within matter almost completely cancel, leaving gravity as the primary force between large bodies such as a person and the Earth. At atomic and molecular scales, however, electrostatic forces are overwhelmingly stronger. They govern how atoms bond to form molecules, how proteins fold into their functional shapes, how ions move across cell membranes, and how electrical signals propagate through nerves and muscle cells.

Section Summary

  • Coulomb's law describes the magnitude of the electrostatic force between two point charges:
[latex]F=k\frac{|q_1q_2|}{r^2}[/latex]
  • The electrostatic force acts along the line joining two charges. Like charges repel, while unlike charges attract.
  • The magnitude of the electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
  • Coulomb's law follows an inverse-square relationship, similar to Newton's law of gravitation.
  • Electrostatic forces govern interactions between atoms, molecules, and ions, making them essential for understanding chemistry, biology, and many medical technologies.
  • At atomic and molecular scales, the electrostatic force is enormously stronger than gravity. Gravity dominates only for large objects because most matter is electrically neutral overall.

Conceptual Questions

  1. Figure 4.3 shows the charge distribution in a water molecule, which is called a polar molecule because it has an inherent separation of charge. Given water's polar character, explain what effect humidity has on removing excess charge from objects.
    Diagram of a polar water molecule showing two slightly positive hydrogen atoms and a slightly negative oxygen atom, with the electron cloud concentrated more near the oxygen.
    Figure 4.3: Schematic representation of the outer electron cloud of a neutral water molecule. The electrons spend more time near the oxygen than the hydrogens, giving water a permanent charge separation. Water is therefore a polar molecule and is more easily affected by electrostatic forces than molecules with uniform charge distributions.
  2. Using Figure 4.3, explain in terms of Coulomb's law why a polar molecule is attracted by both positive and negative charges.
  3. Given the polar character of water molecules, explain how ions in the air form nucleation centers for rain droplets.

Problems & Exercises

  1. What is the repulsive force between two pith balls that are [latex]8.00\,\text{cm}[/latex] apart and have equal charges of [latex]-30.0\,\text{nC}[/latex]?
    1. How strong is the attractive force between a glass rod with a [latex]0.700\,\mu\text{C}[/latex] charge and a silk cloth with a [latex]-0.600\,\mu\text{C}[/latex] charge, which are [latex]12.0\,\text{cm}[/latex] apart, using the approximation that they act like point charges?
    2. Discuss how the answer to this problem might be affected if the charges are distributed over some area and do not act like point charges.
  2. Two point charges exert a [latex]5.00\,\text{N}[/latex] force on each other. What will the force become if the distance between them is increased by a factor of three?
  3. Two point charges are brought closer together, increasing the force between them by a factor of 25. By what factor was their separation decreased?
  4. How far apart must two point charges of [latex]75.0\,\text{nC}[/latex], typical of static electricity, be to have a force of [latex]1.00\,\text{N}[/latex] between them?
  5. If two equal charges of [latex]1.00\,\text{C}[/latex] each are separated in air by a distance of [latex]1.00\,\text{km}[/latex], what is the magnitude of the force acting between them? You will see that even at a distance as large as [latex]1.00\,\text{km}[/latex], the repulsive force is substantial because [latex]1\,\text{C}[/latex] is a very significant amount of charge.
  6. A test charge of [latex]+2\,\mu\text{C}[/latex] is placed halfway between a charge of [latex]+6\,\mu\text{C}[/latex] and another of [latex]+4\,\mu\text{C}[/latex] separated by [latex]10\,\text{cm}[/latex].
    1. What is the magnitude of the force on the test charge?
    2. What is the direction of this force: away from or toward the [latex]+6\,\mu\text{C}[/latex] charge?
  7. Bare free charges do not remain stationary when close together. To illustrate this, calculate the acceleration of two isolated protons separated by [latex]2.00\,\text{nm}[/latex], a typical distance between gas atoms. Explicitly show how you follow the steps in the problem-solving strategy for electrostatics.
    1. By what factor must you change the distance between two point charges to change the force between them by a factor of 10?
    2. Explain how the distance can either increase or decrease by this factor and still cause a factor of 10 change in the force.
  8. Suppose you have a total charge [latex]q_{\text{tot}}[/latex] that you can split in any manner. Once split, the separation distance is fixed. How do you split the charge to achieve the greatest force?
    1. Common transparent tape becomes charged when pulled from a dispenser. If one piece is placed above another, the repulsive force can be great enough to support the top piece's weight. Assuming equal point charges, which is only an approximation, calculate the magnitude of the charge if the electrostatic force is great enough to support the weight of a [latex]10.0\,\text{mg}[/latex] piece of tape held [latex]1.00\,\text{cm}[/latex] above another.
    2. Discuss whether the magnitude of this charge is consistent with what is typical of static electricity.
  9. Find the ratio of the electrostatic force to the gravitational force between two electrons.
    1. What is this ratio for two protons?
    2. Why is the ratio different for electrons and protons?
  10. At what distance is the electrostatic force between two protons equal to the weight of one proton?
  11. A certain five-cent coin contains [latex]5.00\,\text{g}[/latex] of nickel. What fraction of the nickel atoms' electrons, removed and placed [latex]1.00\,\text{m}[/latex] above it, would support the weight of this coin? The atomic mass of nickel is 58.7, and each nickel atom contains 28 electrons and 28 protons.
    1. Two point charges totaling [latex]8.00\,\mu\text{C}[/latex] exert a repulsive force of [latex]0.150\,\text{N}[/latex] on one another when separated by [latex]0.500\,\text{m}[/latex]. What is the charge on each?
    2. What is the charge on each if the force is attractive?
  12. Point charges of [latex]5.00\,\mu\text{C}[/latex] and [latex]-3.00\,\mu\text{C}[/latex] are placed [latex]0.250\,\text{m}[/latex] apart.
    1. Where can a third charge be placed so that the net force on it is zero?
    2. What if both charges are positive?
  13. Two point charges [latex]q_1[/latex] and [latex]q_2[/latex] are [latex]3.00\,\text{m}[/latex] apart, and their total charge is [latex]20\,\mu\text{C}[/latex].
    1. If the force of repulsion between them is [latex]0.075\,\text{N}[/latex], what are magnitudes of the two charges?
    2. If one charge attracts the other with a force of [latex]0.525\,\text{N}[/latex], what are the magnitudes of the two charges? Note that you may need to solve a quadratic equation to reach your answer.

Glossary

Coulomb's law
the physical law that describes the magnitude of the electrostatic force between two point charges
Coulomb force
another name for the electrostatic force between charged objects
electrostatic force
the attractive or repulsive force between electrically charged objects
definition

License

Icon for the Creative Commons Attribution 4.0 International License

Introductory Physics for the Health and Life Sciences II Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Share This Book