Electric Charge and Electric Field

5 Electric Field: Concept of a Field Revisited

Learning Objectives

  • Describe a force field and calculate the strength of an electric field due to a point charge.
  • Calculate the force exerted on a test charge by an electric field.
  • Explain the relationship between electrical force [latex]\mathbf{F}[/latex] on a test charge and electric field strength [latex]\mathbf{E}[/latex].

Contact forces, such as those between a baseball and a bat, are explained at the microscopic level by interactions between charged particles in atoms and molecules. These interactions arise primarily from the Coulomb force. However, not all forces require physical contact. Many forces act over a distance, meaning the interacting objects are separated by space rather than touching. Such interactions are called action-at-a-distance forces.

For example, a charged rubber comb can attract neutral bits of paper from several centimeters away. In biological systems, similar effects occur continuously: ions in extracellular fluid influence nearby ions, charged proteins interact across short distances, and electric signals in nerves propagate through fields in surrounding tissues.

To describe such interactions, physicists use the idea of a field. A field describes how an object influences the space around it, allowing it to exert forces on other objects located at a distance.

Concept of a Field

A field is a way of visualizing and calculating how forces act in space. Instead of thinking of objects somehow “reaching out” to one another, we imagine that an object creates a field in the space around it. Another object placed in that region experiences a force due to the field at its location.

A familiar example is the gravitational field surrounding Earth. Any mass placed near Earth experiences a gravitational force due to this field. In a similar way, every electric charge produces an electric field that fills the surrounding space.

The Coulomb force field surrounding a charge extends outward in all directions. Using Coulomb’s law, the force between a point charge [latex]Q[/latex] and a test charge [latex]q[/latex] separated by a distance [latex]r[/latex] is given by

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

This force depends on both the source charge [latex]Q[/latex] and the test charge [latex]q[/latex]. Figure 5.1 illustrates how the force differs depending on the sign and magnitude of the test charge.

Diagram showing a positive source charge exerting forces on two different test charges, with force direction depending on the sign of the test charge and force magnitude depending on the test charge size.
Figure 5.1: The Coulomb force field due to a positive charge [latex]Q[/latex] acting on two different test charges. The direction and magnitude of force depend on the sign and size of the test charge.

Because the force depends on the test charge, it is useful to define a quantity that describes the influence of the source charge alone. This leads to the definition of the electric field.

The electric field [latex]\mathbf{E}[/latex] is defined as the force per unit charge:

[latex]\mathbf{E}=\frac{\mathbf{F}}{q}[/latex]

This definition means that the electric field describes how strongly a charge would be pushed or pulled if placed at a particular location. The direction of the electric field is defined as the direction of the force on a positive test charge.

It is assumed that the test charge is very small so that it does not disturb the field being measured. The units of electric field are newtons per coulomb (N/C).

If the electric field is known, the force on any charge can be found using

[latex]\mathbf{F}=q\mathbf{E}[/latex]

This relationship is extremely important in biology and medicine. For example, ions such as sodium (Na+), potassium (K+), and calcium (Ca2+) move through tissues and across cell membranes under the influence of electric fields. The movement of these charged particles is essential for nerve signaling, muscle contraction, and cardiac rhythms.

For a point charge [latex]Q[/latex], combining Coulomb’s law with the definition of electric field gives

[latex]E=\frac{F}{q}=k\frac{|Q|}{r^2}[/latex]

This result shows that the electric field depends only on the source charge and the distance from it, not on the test charge used to measure it.

The inverse-square dependence means that electric fields decrease rapidly with distance. This is important in physiology, where electric effects between ions are strongest at nanometer scales but weaken quickly as distance increases.

Example 5.1: Calculating the Electric Field of a Point Charge

Calculate the magnitude and direction of the electric field produced by a point charge of [latex]2.00\ \text{nC}[/latex] at a distance of [latex]5.00\ \text{mm}[/latex] from the charge.

Strategy

Since the source is a point charge, we use the equation for the electric field of a point charge:

[latex]E=k\frac{|Q|}{r^2}.[/latex]

First convert all quantities to SI units, then substitute the known values into the equation.

Solution

The given values are

[latex]Q=2.00\times10^{-9}\ \text{C}[/latex]

and

[latex]r=5.00\times10^{-3}\ \text{m}.[/latex]

Substituting into the electric field equation gives

[latex]\begin{aligned} E&=k\frac{|Q|}{r^2} \\ &=\left(8.99\times10^9\ \frac{\text{N}\cdot\text{m}^2}{\text{C}^2}\right) \frac{2.00\times10^{-9}\ \text{C}} {(5.00\times10^{-3}\ \text{m})^2} \\ &=7.19\times10^5\ \text{N/C}. \end{aligned}[/latex]

Because the source charge is positive, the electric field points away from the charge.

Discussion

The magnitude of the electric field depends only on the source charge and the distance from it. Every point located [latex]5.00\ \text{mm}[/latex] from this charge has the same field strength, [latex]7.19\times10^5\ \text{N/C}[/latex], although the direction changes from point to point and always points radially away from the positive charge. If the source charge were negative instead, the field would have the same magnitude but would point toward the charge.

Example 5.2: Calculating the Force Exerted on a Charge by an Electric Field

What force does the electric field found in Example 5.1 exert on a point charge of [latex]-0.250\ \mu\text{C}[/latex]?

Strategy

Since the electric field strength is known, we can calculate the force using the relationship

[latex]\mathbf{F}=q\mathbf{E}.[/latex]

The sign of the charge determines whether the force points in the same direction as the field or in the opposite direction.

Solution

Substitute the known values into the equation:

[latex]F=qE[/latex]
[latex]F=\left(-0.250\times10^{-6}\ \text{C}\right)\left(7.19\times10^{5}\ \text{N/C}\right)[/latex]
[latex]F=-1.80\times10^{-1}\ \text{N}=-0.180\ \text{N}[/latex]

The negative sign indicates that the force is directed opposite to the electric field.

Discussion

Because the electric field was produced by a positive source charge, it points away from that charge. A negative test charge experiences a force in the opposite direction, so it is attracted toward the positive charge. This example illustrates an important rule: positive charges move in the direction of the electric field, whereas negative charges move opposite to it.

Interactive Exploration: Electric Field of Dreams

Electric fields describe how electric charges influence the space around them. Any charged particle placed within an electric field experiences an electric force, causing it to accelerate. In this simulation, you will investigate how positive and negative charges respond to electric fields created by other charges as well as by a uniform external field.

Experiment by placing charges on the field and observing their motion. Change the magnitude and direction of the background electric field, and compare how positive and negative charges respond. As you explore, think about how electric field lines indicate both the direction and strength of the electric force acting on a charged particle.

Figure 5.2: In this interactive PhET simulation, positive and negative charges move in response to electric fields. Positive charges accelerate in the direction of the electric field, while negative charges accelerate in the opposite direction.

Accessibility note: If you are unable to use the interactive simulation, read the guided exploration questions first and compare your predictions with the discussion that follows. Your instructor may also provide screenshots or a demonstration of the simulation.

Guided Exploration

As you interact with the simulation, try to answer the following questions:

  1. Place a single positive charge in the simulation. How does it respond when you turn on a uniform electric field?
  2. Replace the positive charge with a negative charge. How does its motion compare with that of the positive charge?
  3. Increase the strength of the background electric field. How does this affect the force and acceleration experienced by the charges?
  4. Place multiple charges in the field. How do the electric forces from nearby charges combine with the background electric field?
  5. Reverse the direction of the electric field. What changes do you observe in the motion of positive and negative charges?
  6. Based on your observations, explain how the direction of an electric field is defined and why positive and negative charges move in opposite directions.

After completing the exploration, compare your observations with the concepts presented in this chapter. Notice that the electric field determines the direction of the force on a positive test charge. Positive charges accelerate in the direction of the field, while negative charges accelerate in the opposite direction. The stronger the electric field, the greater the electric force exerted on a charged particle.

Section Summary

  • An electric field describes how a charged object influences the space around it and exerts forces on other charges at a distance.
  • The electric field [latex]\mathbf{E}[/latex] is defined as the electric force per unit positive test charge:
[latex]\mathbf{E}=\frac{\mathbf{F}}{q}[/latex]
  • The SI unit of electric field is newtons per coulomb (N/C).
  • If the electric field is known, the force on a charge is found from:
[latex]\mathbf{F}=q\mathbf{E}[/latex]
  • The magnitude of the electric field created by a point charge [latex]Q[/latex] at distance [latex]r[/latex] is:
[latex]E=k\frac{|Q|}{r^2}[/latex]
  • The electric field is a vector. The field points away from positive charges and toward negative charges.
  • Electric fields from multiple charges add as vectors.

Conceptual Questions

  1. Why must the test charge [latex]q[/latex] in the definition of the electric field be vanishingly small?
  2. Are the direction and magnitude of the Coulomb force unique at a given point in space? What about the electric field?

Problems & Exercises

  1. What is the magnitude and direction of an electric field that exerts a [latex]2.00\times10^{-5}\ \text{N}[/latex] upward force on a [latex]-1.75\ \mu\text{C}[/latex] charge?
  2. What is the magnitude and direction of the force exerted on a [latex]3.50\ \mu\text{C}[/latex] charge by a [latex]250\ \text{N/C}[/latex] electric field that points due east?
  3. Calculate the magnitude of the electric field [latex]2.00\ \text{m}[/latex] from a point charge of [latex]5.00\ \text{mC}[/latex], such as found on the terminal of a Van de Graaff generator.
    1. What magnitude point charge creates a [latex]10{,}000\ \text{N/C}[/latex] electric field at a distance of [latex]0.250\ \text{m}[/latex]?
    2. How large is the field at [latex]10.0\ \text{m}[/latex]?
  4. Calculate the initial acceleration from rest of a proton in a [latex]5.00\times10^{6}\ \text{N/C}[/latex] electric field, such as one created by a research Van de Graaff generator. Explicitly show how you follow the steps in the problem-solving strategy for electrostatics.
    1. Find the direction and magnitude of an electric field that exerts a [latex]4.80\times10^{-17}\ \text{N}[/latex] westward force on an electron.
    2. What magnitude and direction force does this field exert on a proton?

Glossary

field
a physical quantity that describes how an object influences the space around it, allowing it to exert forces on other objects at a distance
point charge
an idealized charged object whose size is negligible compared with the distances involved, allowing it to be treated as a single point
test charge
a very small positive charge used to measure an electric field without significantly disturbing it
electric field
a vector field that gives the electric force per unit positive test charge at each point in space
definition

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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.

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