Electricity and mathematics

Summary: Electricity, arising from the flow of electrons, can be described mathematically.

Daily operations of modern industrial societies, including transportation, communication, heating, cooling, lighting, computing, and medical technology, rely on the use of electrical power. Power from batteries and electrical outlets is derived from the flow of electrons, known as “electric current.” The term “electricity” refers to a variety of physical effects, both static and dynamic, that arise from electric charge. The mathematical description of electric and magnetic phenomena developed in the eighteenth and nineteenth centuries contributed to a rapid expansion of electrical technology, which is powered today by a vast grid of electric power stations and distribution systems.

Electric Charge and Coulomb’s Law

Electric charge is a property of matter that can be negative (as in electrons), positive (as in protons), or zero. Most matter has a net charge of zero, containing essentially the same number of electrons as protons. Two objects whose charges are both positive or both negative repel each other, while objects with opposite charges attract each other. Static electricity is created when electrons build up on or are depleted from the surface of a material, often by rubbing materials together. Effects of static electricity are seen, for example, in a rubbed balloon clinging to a wall, or in hair standing on end. In metals, electrons are not strongly bound to individual atoms but move freely through the lattice of protons. Materials with freely moving charges are known as “conductors.” The force between two charged particles at rest is described by Coulomb’s Law, named after French engineer Charles-Augustin de Coulomb (1736-1806). Coulomb’s Law states that the magnitude f of the force exerted by one charged particle on the other is

where q and q′ are the magnitudes of the charges of the particles, r is the distance between the two particles, and k is a constant. This equation shows, for example, that if one charge is tripled, then the force is tripled, and if both charges are tripled, then the force becomes nine times as large. On the other hand, tripling the distance r between the particles multiplies the right-hand side of the equation by 1/3², or 1/9, reducing the force to a ninth of its previous value.

Electric Field and Electric Current

The presence of charged particles creates an electric field that exerts a force on other charged particles in the region. An electric power generator, usually driven by a steam turbine fueled by coal or a nuclear reactor, creates an electric field between two terminals by building an over-supply of electrons (negative charge) in one terminal and a deficit of electrons (positive charge) in the other. The flow of electrons from a negative toward a positive terminal along a conducting path, such as a wire, is an electric current. In lightning, electrons from negatively charged clouds in the atmosphere are attracted to positively charged objects on the ground beneath the cloud. Here the electric field is so strong that electric current passes through air, which usually acts as an insulator that prevents the flow of electrons.Batteries operate by producing an electric current between oppositely charged terminals of chemical cells. A battery produces direct current (DC), where electrons flow in one direction, while a power generator creates alternating current (AC), where the direction of electron flow alternates rapidly, typically at a frequency of 60 hertz (cycles per second). The hertz is named for German physicist Heinrich Hertz (1857-1894), who made important advances in understanding the connection between electric and magnetic fields.

Ohm’s Law

The energy that an electric field imparts to a unit charge moving from one terminal to another is the number of volts (V) between the terminals, named after Italian physicist Alessandro Volta (1745-1827). On electric bills, energy usage is typically given in kilowatt hours (kWh). The watt, named for British engineer James Watt (1736-1819), is a unit of power, or energy per time, and 1 kilowatt is 1000 watts. Multiplying power (in kilowatts) by time (in hours) yields energy, in kilowatt-hours. In an electric current, the current intensity (I) is abbreviated as “current” and is the quantity of charge that moves past a cross-section of the conducting path per unit time. As electric current flows through a material, the motion of the electrons is hindered by positive ions, creating electrical resistance (R). Resistance in the path of a current creates heat and light, as in appliances, such as stoves and light bulbs. Electrical energy can be transformed into mechanical energy to power motors as in cars, airplanes, power tools, kitchen blenders, and hair dryers when electric current passes through a coil of wire, inducing a magnetic field that sets the coil in motion.

Ohms’s Law, formulated by German physicist Georg Ohm (1789-1854), states that for a metal conductor at constant temperature, the voltage (V) is V=IR, where I is the current, and R is the resistance. This equation shows, for example, that if the resistance is cut in half, then to maintain the same voltage, the current must be doubled. If too little resistance is present, the current may become so strong as to damage electrical equipment. Circuit breakers then sever the path of the current to avoid damage.

Electric Power from Generator to Consumer

High voltage generated at power stations is propagated along power lines almost instantaneously, over many miles, to substations near cities and towns. At the substations, the voltage is reduced and transmitted to electric distribution centers that channel the voltage to homes, offices, and other facilities. In standard electrical outlets in the United States, there are 120 volts between the wires leading to the two vertical slots. When an appliance is plugged into the outlet, the vertical prongs of the plug make contact with these wires, creating a pathway of current through the appliance. The third slot in the outlet carries a protective ground wire. In appliances with a three-pronged plug, the ground wiring is designed to provide a preferred pathway for escaped current so that it will not travel through the body of the person holding the appliance.

Large appliances, including most drying machines and ovens, operate at 240 volts, using a different type of outlet. Touching one or more openings in an electrical outlet or touching the prongs of a plug as it is inserted into the outlet may pass an electric current through the body that can be harmful or even deadly. At electrical facilities, “High Voltage” signs warn of the danger of electric shock because of the presence of high voltage.

Bibliography

California Energy Commission. “What Is Electricity?” http://energyquest.ca.gov/story/chapter02.html.

Herman, Stephen L., and Crawford G. Garrard. Practical Problems in Mathematics for Electricians. 6th ed. Albany, NY: Delmar, 2002.

U.S. Energy Information Administration: Independent Statistics and Analysis. “Electricity.” http://www.eia.doe.gov/fuelelectric.html.