Forces On Charges And Currents

Type of physical science: Classical physics

Field of study: Electromagnetism

The electromagnetic force, one of the four known basic forces in nature, results from the interaction between charges and currents. The electric force is an interaction between charges and the magnetic force is an interaction between currents.

Overview

Fundamentally, the electric force is an interaction between charges and the magnetic force is an interaction between currents. These forces are often described using the concepts of electric and magnetic fields.

Charge is either positive or negative. Positive charges are repelled by other positive charges. Negative charges are repelled by other negative charges. Yet, positive charges and negative charges are attracted to each other. Charge is a basic property of matter that must be conserved in all interactions and reactions. Charge does not exist in naked form; it is always associated with matter. Charge is also quantized; in other words, it exists only in integral multiples of the elementary charge of nature (e = 1.6 x 10^-19 coulombs), which is the magnitude of charge of the electron or the proton. Electrons are the principal carriers of negative charge and protons are the principal carriers of positive charge.

Charge in motion constitutes a current. Matter can be categorized in terms of how well it conducts a flow of current. In a conductor, the outermost atomic electrons are relatively free to move throughout the material under the influence of a periodic electric potential or ionic cores. Many metals, such as copper, silver, and gold, are excellent conductors of electricity. In an insulator, the outermost atomic electrons are tightly bound to individual atoms. Although no perfect insulator exists, many materials, such as quartz, conduct electricity 10²⁵ times less effectively than metallic conductors. Semiconductors have an intermediate ability to conduct current. Using a modification process known as doping, semiconductor behavior forms the basis of most modern solid-state devices, such as diodes and transistors. Because the ability to conduct both electrical current and thermal energy is heavily dependent upon atomic structure, good electrical conductors are also good conductors of heat, and good electrical insulators are also good thermal insulators.

If two particles (q₁ and q₂) are charged, their force of interaction is described by Coulomb's law of electrostatics. That force is proportional to the magnitude of each charge and is inversely proportional to the square of the distance separating the charges. This behavior is similar to the gravitational interaction between two masses, as described by Sir Isaac Newton's law of universal gravitation; whereas gravitation is always attractive, however, the Coulomb electrostatic force can either be attractive (+ and -) or repulsive (+ and + or - and -). Like gravitation, the electrostatic force is directed along the line joining the centers of the two charged particles. Like charges are repelled and separate along the line joining their centers, and opposite charges are attracted along the line joining their centers.

The constant of proportionality in the Coulomb law involves a fundamental constant of nature called the permittivity of free space. Its value helps to define the speed of electromagnetic radiation in a vacuum and the strength of the electrostatic interaction relative to the gravitational interaction. In the Coulomb law of electrostatic interaction, the full constant of proportionality (k) has a value of 9 x 10⁹ newtons x meters squared per Coulomb squared.

The electric force is a vector quantity, having both a magnitude and a direction. Thus, if several charges are present in a given region of space, the net force on any single charge is the vector sum of forces caused by the presence of other charges acting on the charge in question. For example, consider four charges q₁, q₂, q₃, and q₄. To find the force on charge q₁, the distance between q₁ and q₂ (r₁₂), the distance between q₁ and q₃ (r₁₃), and the distance between q₁ and q₄ (r₁₄) must be known. The force on charge q₁ caused by q₂ is F₁₂, the force on charge q₁ caused by q₃ is F₁₃, and the force on charge q₁ caused by q₄ is F₁₄. Force F₁₂ has a value kq₁q₂/r₁₂², force F₁₃ has a value kq₁q₃/r₁₃², and force F₁₄ has a value kq₁q₄/r₁₄². These forces are attractive if the product of qⁱq^j (i≠j) is negative and repulsive if the product is positive. These forces do not act along the same line; as a result, the net force is not simply F₁₂ + F₁₃ + F₁₄. The directions in which F₁₂, F₁₃, and F₁₄ act must be taken into account when these forces are added together vectorially.

In describing the electrostatic force, the concept of "lines of force" is often utilized. Lines of force are drawn radially outward from positive charges and radially inward toward negative charges. Lines of force indicate the direction a positive test charge would move in response to the force exerted on it by a charge in question. In drawing lines of force, the density of lines is meant to indicate qualitatively the relative strength of the force: where the density of lines is high, the force is strong; where the density is low, the force is weak.

Consider an extended object that is charged. If a "point charge" is brought into the presence of this extended object, what force of electrostatic interaction will result? The approach used to determine this force is to break up the charge distributed on or in the extended object into small elements of charge that can be considered as point charges and then describe the interaction between each small element and the exterior point charge. The smaller the element of charge, the greater the number of elements. The vector sum of a large number of interactions can become cumbersome. The larger the number of elements, the closer the approximation to the true force of interaction. As the number of small elements becomes infinite, the vector sum of interactions becomes an integral over the charge distribution. A knowledge of the charge density distribution within and on the extended object is required to evaluate this integral.

A charge q moving with speed v into a region of magnetic field B experiences a magnetic force given by F = qv B sin phi, where phi is the angle between the direction of the velocity vector and the magnetic field. The direction of this force is determined by the right-hand rule. (To make use of the right-hand rule, point the fingers in the direction of the velocity and turn them toward the magnetic field vector. The extended thumb points in the direction of the magnetic force.) Thus, the velocity, magnetic field, and magnetic force form a triad of mutually perpendicular vectors. A magnetic force accelerates a charged particle in a direction perpendicular to both the magnetic field and the direction of motion. In contrast, an electric force accelerates a charged particle along the direction of the electric field and direction of motion.

The magnetic field B does not work on a charged particle q moving at speed v during a displacement dL. Work is the scalar product of force and displacement vectors, meaning the multiplication of the component of force along the direction of the displacement by the displacement itself. Because the magnetic force is perpendicular to the velocity vector, which shares the same direction as the displacement, there is no component for force along dL. By the work-energy theorem, it is seen that magnetic forces, which perform no work on moving charged particles, do not change the kinetic energy of charged particles moving in a magnetic field.

If a charged particle enters a region of space where a magnetic field exists, the particle will execute a helical motion. The velocity of the charged particle could be analyzed into a component parallel to the field (u) and a component perpendicular to the field (w). Since the magnetic force acting on the charged particle is perpendicular to both the velocity and magnetic field, only the component of the velocity perpendicular to the field (w) is involved in the calculation of the magnetic force. There will not be an acceleration of the charged particle in the direction of the velocity component parallel to the magnetic field. Therefore, the charged particle will continue to move in the direction of the magnetic field with a speed given by u, while executing a circular motion in a plane perpendicular to the magnetic field and velocity vector as a result of the sideways deflecting force determined by qwB. This circular motion in the plane results because the acceleration is centripetal in nature.

If a charged particle enters a region of space where there exists both an electric field E and magnetic field B, the total force on the charged particle, referred to as the Lorentz force, is given as the vector sum of the electric force and magnetic force given by F_net = qE + qv × B, where E, B, and F are vectors and the x indicates a vector cross product. The electric force accelerates the charge in the direction of the electric field, and the magnetic force accelerates the particle in the direction perpendicular to both the velocity vector and magnetic field.

When a current i flows through a conducting wire, it creates a magnetic field. If that wire is immersed in an external magnetic field B , then the force acting on the wire is given by F = i LB sin phi, where L is the wire length (actually a vector of magnitude equal to the wire's length with a direction dictated by the direction of current flow through the wire) and phi is the angle between B and L . This force will be perpendicular to both B and L.

Applications

In the case of the hydrogen atom, a single electron orbits around a single proton. The electric force of attraction between these two opposite but equally charged particles is centripetally directed. That force is -ke ²/r ² and can be set equal to -mv ²/r. From this relationship, the speed of the electron in one of its quantized orbits and hence its energy can be calculated, such as in the ground state r = 5.3 x 10^-11 meters. This yields an electron speed of 2.2 x 10⁶ meters per second and electron energy of -13.6 electronvolts. To ionize the hydrogen atom (remove the electron from its bound state), an energy of +13.6 electronvolts must be added to the system.

Consider the nucleus of an iron atom. Inside an iron atom's nucleus are twenty-six positively charged protons that are separated by approximately 4 fermi (10^-15 meters). How does the repulsive electrostatic force between any pair of protons compare with their mutual gravitational attraction? By Coulomb's law, the electrostatic repulsion is 14.4 newtons, and by Newton's law of universal gravitation, the gravitational attraction is 1.2 x 10^-35 newtons. One can see that gravity cannot hold the nucleus together. It is the strong nuclear force that is responsible for binding nucleons together inside the nucleus. This example demonstrates the strength of the electrostatic interaction relative to the gravitational interaction; the electrostatic force is thirty-six orders of magnitude stronger than the gravitational force.

Electrostatic forces and gravitational forces share several characteristics in common. Both forces are long-ranged, going only to zero as distance between interacting particles goes to infinity. Both forces obey inverse square law distance dependence. Both forces are directly proportional to the property that is responsible for the interaction, charge for electrostatics and mass for gravitation.

Inside a conducting material, charges are free to move under the influence of the electrostatic interaction. As a result, positive charges will be repelled by other positive charges, negative charges will be repelled by other negative charges, and negative charges will be attracted to positive charges. This motion will continue until equilibrium is established by maximizing the positive-positive and negative-negative charge distance and minimizing the positive-negative charge distance. All free charge inside a conductor rapidly moves to the surface of the conductor to establish equilibrium where there is no net force on any individual charge. This represents the electrostatic condition. A nonelectrostatic condition can be set up inside the conductor by connecting it to a source of electrostatic potential. By placing a potential difference across the conductor, positive charges will move from high electric potential to low potential, and negative charges will move from low electric potential to high potential. Both motions result in lower electrical potential energy. This condition represents a flow of current through the conductor limited only by the material's resistance.

Suppose a 10-centimeter length of wire carries a 5-ampere current. The wire is immersed in a magnetic field of strength 2 millitesla that is oriented at a 30 degree angle to the wire. Since F = i LB sin phi, this force has a magnitude of 5 x 10^-4 newtons. If the wire and magnetic field were in the plane of this page, then the magnetic force acting on the wire would be oriented either perpendicularly out of the page if the current was oriented clockwise from B or perpendicularly into the page if the current direction was oriented counterclockwise from B .

Current traveling through each of two parallel wires will create magnetic fields that are sensed by the other wire. Current is the cause and magnetic field (hence magnetic force) is the effect. The magnetic field of one wire serves as an external field for the other and thus these current carrying wires will exert forces on each other. The force on wire 1 will be given by i₁L₁B₂ where i₁ and L₁ are the current and length of wire 1, respectively, and B₂ is the strength of the magnetic field generated by wire 2 sensed at the location of wire 1. The force on wire 2 will be given by i₂L₂B₁, where i₂ and L₂ are the current and length of wire 2, respectively, and B₁ is the strength of the magnetic field generated by wire 1 sensed at the location of wire 2. If these parallel wires carry currents flowing in the same direction, these forces will be attractive. If these parallel wires carry currents flowing in opposite directions, these forces will be repulsive.

To determine exactly how large such forces are, suppose a current of 100 milliamperes flows through a pair of 10-centimeter long wires separated by a distance of 10 centimeters. The force of interaction is a mere 2 x 10^-8 newtons.

Context

Electric forces and magnetic forces are fundamental forces of interaction between charged particles. The interaction between charges is the electric force, and the interaction between currents (charge in motion) is the magnetic force. The electric force strength is determined in part by the permittivity of free space, which is a fundamental constant of nature; the magnetic force strength is determined in part by the permeability of free space, which is another fundamental constant of nature. Charge is the underlying physical property of matter that links electricity to magnetism. Indeed, electric and magnetic forces are different manifestations of the same fundamental force of interaction: electromagnetism. Many centuries of study and experimentation followed the discovery of electric and magnetic forces before the unification of electric and magnetic phenomena under the framework of electromagnetism was realized.

There are four known fundamental forces of interaction in nature: gravitation, electromagnetism, the weak nuclear force, and the strong nuclear force. Gravitation is the force of attraction between material particles and all other material particles. It is a weak but long-ranged force. The weak nuclear force is a strong but short-ranged force involved in radioactive decay processes. The strong nuclear force is the strongest, but shortest-ranged force. It is the force that binds nucleons together in a nucleus, despite the electrostatic repulsion between pairs of positively charged protons. Electromagnetism and the weak nuclear force have been successfully unified into the electroweak interaction. A major effort in physics involves an attempt to unify these four fundamental forces of nature into a single framework.

Principal terms:

CHARGE: a basic property of matter, which occurs as either positive (+) or negative (-); opposites attract and likes repel according to Coulomb's law

CONDUCTOR: a material whose atomic structure permits a reasonable flow of charge

CURRENT: the flow of charge; direction is determined by the flow of positive charge

ELECTRIC FIELD: the force per unit charge exerted on a given charge by an interaction with another charge distribution

FORCE: by Newton's second law of motion, the cause of an acceleration resisted by a particle's mass

LORENTZ FORCE: the net force resulting from a combined electric and magnetic field acting on a charged particle

MAGNETIC FIELD: a means of describing the sideways deflecting force experienced by a moving charge in the presence of atomic or macroscopic currents

Bibliography

Frankyl, Daniel R. ELECTROMAGNETIC THEORY. Englewood Cliffs, N.J.: Prentice-Hall, 1986. Although targeted for the undergraduate physics major, this text contains qualitative sections on the significance of electric and magnetic fields, and explains the electromagnetic forces of interation between charged particles in motion.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 3d rev. ed. New York: John Wiley & Sons, 1988. This version of the classic undergraduate physics text contains descriptive discussions of current topics in physics research as well as thorough treatments of basic physical concepts. Excellent descriptions of electromagnetic forces of interaction. Numerous illustrations and sample problems.

Lorrain, Paul, and Dale Corson. ELECTROMAGNETIC FIELDS AND WAVES. New York: W. H. Freeman, 1990. A classic text used for undergraduate instruction in electromagnetism. Many sections are accessible to the layperson with moderate mathematical skills. Descriptions of field concepts are excellent.

Nayfeh, Munir H., and Morton K. Brussel. ELECTRICITY AND MAGNETISM. New York: John Wiley & Sons, 1985. Undergraduate-level instruction in mathematical and experimental aspects of electromagnetism. Accessible to those with modest skills in calculus of vectors.

Ohanian, Hans C. PHYSICS. New York: W. W. Norton, 1985. Although calculus-based, the text is not mathematically rigorous. Accessible to those with modest mathematical skills. Excellent descriptions of difficult physical concepts, with good illustrations and practical examples.

Reitz, John R., and Frederick J. Milford. FOUNDATIONS OF ELECTROMAGNETIC THEORY. Reading, Mass.: Addison-Wesley, 1987. A classic text for undergraduate instruction of electromagnetism. Excellent field concept descriptions. Accessible to the average reader with moderate mathematical skills.

Tomboulian, D. H. ELECTRIC AND MAGNETIC FIELDS. New York: Harcourt, Brace & World, 1965. Although somewhat dated, this text is ideal for the amateur interested in an increased understanding of electromagnetism without requiring higher-level mathematics. Well illustrated, with many diagrams and sample problems.

Wangsness, Roald K. ELECTROMAGNETIC FIELDS. New York: John Wiley & Sons, 1986. Highly mathematical, but descriptions of basic concepts are illuminating. For the seriously interested layperson with good calculus skills.

Wilson, Jerry D., and John Kinard. COLLEGE PHYSICS. Boston: Allyn & Bacon, 1990. Basic text, excellent for the reader not familiar with calculus. More qualitative than rigorous in nature. Well illustrated, with a thorough description of basic concepts without resorting to advanced mathematics. Excellent for high school physics instruction.

Like charges repel each other; opposite charges attract

Charges and Currents