Electric Current, Resistance, and Ohm’s Law
20 Ohm’s Law: Resistance and Simple Circuits
Learning Objectives
- Explain the origin of Ohm’s law.
- Calculate voltage, current, or resistance using Ohm’s law.
- Explain what an ohmic material is.
- Describe the components of a simple electric circuit.
In the previous chapter, we defined electric current as the rate at which electric charge flows. But what causes charges to move? In everyday circuits, the driving force is a potential difference, or voltage, produced by a battery, generator, or electrical outlet. Devices that provide a voltage are called voltage sources.
When a voltage source is connected to a conductor, it establishes an electric field inside the conductor. That electric field exerts forces on the charge carriers, causing them to drift and producing an electric current. In this chapter, we examine the relationship between voltage, current, and the property of materials that opposes current: electrical resistance.
Ohm's Law
Experiments show that, for many materials under ordinary conditions, the current flowing through the material is directly proportional to the voltage applied across it. The German physicist Georg Simon Ohm (1787–1854) carefully measured this relationship using metal wires and found that doubling the applied voltage doubled the current:
This relationship is known as Ohm's law. It is important to recognize that Ohm's law is an empirical law, meaning it is based on experimental observation rather than derived from more fundamental principles. It accurately describes the behavior of many common conductors, especially metals, provided that temperature and other physical conditions remain approximately constant.
Not every material obeys Ohm's law. Devices such as diodes, LEDs, transistors, superconductors, and many biological tissues exhibit a nonlinear relationship between voltage and current. In these materials, doubling the voltage does not necessarily double the current.
Health and Bioscience Connection
Many medical instruments are designed to operate in ranges where electrical components behave approximately ohmically. A predictable relationship between voltage and current makes sensors easier to calibrate and helps ensure safe, reliable operation. Biological materials, however, are often non-ohmic. The electrical properties of skin, muscle, and cell membranes depend on factors such as hydration, temperature, frequency, and electrode contact, so their resistance is not always constant.
Resistance and Simple Circuits
If voltage drives current, what limits how much current flows? The property that opposes the motion of electric charge is called electrical resistance, represented by the symbol [latex]R[/latex]. As charge carriers move through a material, they collide with atoms, ions, and other particles. These collisions impede their motion and transfer energy to the material, often producing heat.
For many materials, increasing the resistance decreases the current. In other words, current is inversely proportional to resistance:
For example, if the resistance doubles while the applied voltage remains the same, the current is reduced by one-half.
Combining the proportionalities
and
gives the familiar equation for Ohm's law:
Materials that obey this equation are called ohmic materials. For an ohmic material, the resistance remains approximately constant as the voltage and current change, provided the temperature and other physical conditions remain unchanged.
A device manufactured to provide a specific resistance is called a resistor. The SI unit of resistance is the ohm, whose symbol is the Greek capital omega, [latex]\Omega[/latex]. Rearranging Ohm's law gives
which defines the unit of resistance:
Figure 20.1 shows a simple electric circuit. It consists of a single voltage source connected to a single resistor by conducting wires. In introductory circuit analysis, the wires are assumed to have negligible resistance, so essentially all of the resistance is provided by the resistor.
Example 20.1: Calculating Resistance: An Automobile Headlight
What is the resistance of an automobile headlight if 2.50 A of current flows through it when a voltage of 12.0 V is applied?
Strategy
Use Ohm's law. Rearrange the equation
to solve for the unknown resistance, then substitute the given values.
Solution
Rearranging Ohm's law gives
Substituting the known values,
Discussion
A resistance of 4.80 Ω is typical for an operating automobile headlight. Notice that this value corresponds to the bulb after it has warmed up. The filament's resistance is lower when it is cold, so immediately after the headlight is switched on, the current is briefly larger. As the filament heats, its resistance increases until it reaches its normal operating value.
Electrical resistances span an enormous range. Some ceramic insulators used to support power lines can have resistances greater than [latex]10^{12}\ \Omega[/latex]. A dry person may have a hand-to-foot resistance of about [latex]10^{5}\ \Omega[/latex], whereas the resistance of the human heart is approximately [latex]10^{3}\ \Omega[/latex]. A one-meter length of thick copper wire may have a resistance of only [latex]10^{-5}\ \Omega[/latex], while superconductors have essentially zero electrical resistance. As we will see later, a material's resistance depends not only on the material itself but also on its size and shape.
Ohm's law can also be written in a form that emphasizes the voltage across a resistor. Solving
for voltage gives
This equation tells us the voltage drop (often called the [latex]IR[/latex] drop) across a resistor carrying a current [latex]I[/latex]. In a circuit, voltage increases across a voltage source and decreases across resistive elements as electrical energy is transferred to other forms.
A useful analogy is water flowing through pipes. A battery acts like a pump that creates a pressure difference, while a resistor behaves like a narrow section of pipe that limits the flow and causes a pressure drop.
This behavior reflects the conservation of energy. The voltage source supplies electrical energy to the moving charges, and the resistor converts that energy into other forms such as heat or light. In a simple circuit containing only one resistor, the entire voltage supplied by the battery appears across that resistor, as illustrated in Figure 20.2.
Making Connections: Conservation of Energy
In a simple circuit, the voltage source supplies electrical energy to the moving charges, while the resistor converts that energy into other forms. For an ordinary resistor, most of the electrical energy becomes thermal energy, although it may also produce light (as in an incandescent bulb) or perform other useful functions. Conservation of energy requires that all of the energy delivered by the source be accounted for by these energy transformations.
Interactive Exploration: Ohm's Law
Ohm's law describes the relationship between voltage, current, and resistance in a simple electric circuit. In this simulation, you'll investigate how changing the voltage or resistance affects the current through a resistor. This will help you connect the equation [latex]I = \frac{V}{R}[/latex] to the behavior of an actual circuit.
Experiment by adjusting the voltage and resistance sliders. Watch how the current changes and how the symbols in the equation respond. As you explore, think about voltage as the electrical push, resistance as the opposition to current, and current as the flow of electric charge.
Guided Exploration
As you interact with the simulation, try to answer the following questions:
- Increase the voltage while keeping resistance constant. What happens to the current?
- Increase the resistance while keeping voltage constant. What happens to the current?
- Set a large voltage and a small resistance. What happens to the current? Why?
- Set a small voltage and a large resistance. What happens to the current? Why?
- Observe the equation display. How do the changing symbol sizes help represent the relationship among voltage, current, and resistance?
- Based on your observations, explain Ohm's law in words without using the equation.
After completing the exploration, compare your observations with the concepts presented in this chapter. Notice that current increases when voltage increases, but decreases when resistance increases. Ohm's law gives a simple mathematical model for predicting how charge flows through many electrical circuits.
Section Summary
- A simple circuit consists of a single voltage source connected to a single resistor (load) by conducting wires.
- For ohmic materials, current, voltage, and resistance are related by Ohm's law:
[latex]I=\frac{V}{R}[/latex]
- The SI unit of resistance is the ohm ([latex]\Omega[/latex]):
[latex]1~\Omega=1~\frac{\text{V}}{\text{A}}[/latex]
- Rearranging Ohm's law gives the voltage drop across a resistor:
[latex]V=IR[/latex]
This voltage drop represents the electrical energy transferred to the resistor per unit charge.
Conceptual Questions
- The [latex]\text{IR}[/latex] drop across a resistor means that there is a change in potential or voltage across the resistor. Is there any change in current as it passes through a resistor? Explain.
- How is the [latex]\text{IR}[/latex] drop in a resistor similar to the pressure drop in a fluid flowing through a pipe?
Problems & Exercises
- What current flows through the bulb of a 3.00-V flashlight when its hot resistance is [latex]3\text{.}\text{60 Ω}[/latex]?
- Calculate the effective resistance of a pocket calculator that has a 1.35-V battery and through which 0.200 mA flows.
- What is the effective resistance of a car’s starter motor when 150 A flows through it as the car battery applies 11.0 V to the motor?
- How many volts are supplied to operate an indicator light on a DVD player that has a resistance of [latex]1\text{40}\phantom{\rule{0.25em}{0ex}}\Omega[/latex], given that 25.0 mA passes through it
- (a) Find the voltage drop in an extension cord having a [latex]0\text{.}\text{0600-}\Omega[/latex] resistance and through which 5.00 A is flowing. (b) A cheaper cord utilizes thinner wire and has a resistance of [latex]0\text{.}\text{300}\phantom{\rule{0.25em}{0ex}}\Omega[/latex]. What is the voltage drop in it when 5.00 A flows? (c) Why is the voltage to whatever appliance is being used reduced by this amount? What is the effect on the appliance?
- A power transmission line is hung from metal towers with glass insulators having a resistance of [latex]1\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\Omega .[/latex] What current flows through the insulator if the voltage is 200 kV? (Some high-voltage lines are DC.)
Glossary
- Ohm's law
- An empirical relationship stating that, for an ohmic material, the current is directly proportional to the applied voltage; commonly written as [latex]I=\frac{V}{R}[/latex].
- resistance
- The electrical property of a material that opposes the flow of electric current; for an ohmic material, [latex]R=\frac{V}{I}[/latex].
- ohm
- The SI unit of electrical resistance, equal to one volt per ampere: [latex]1~\Omega=1~\frac{\text{V}}{\text{A}}[/latex].
- ohmic material
- A material whose current is directly proportional to the applied voltage, so that it obeys Ohm's law over its operating range.
- simple circuit
- A circuit consisting of a single voltage source connected to a single resistor (load).
An empirical relationship stating that, for an ohmic material, the current is directly proportional to the applied voltage; commonly written as [latex]I=\frac{V}{R}[/latex].
The electrical property of a material that opposes the flow of electric current; for an ohmic material, [latex]R=\frac{V}{I}[/latex].
The SI unit of electrical resistance, equal to one volt per ampere: [latex]1~\Omega=1~\frac{\text{V}}{\text{A}}[/latex].
A material whose current is directly proportional to the applied voltage, so that it obeys Ohm's law over its operating range.
A circuit consisting of a single voltage source connected to a single resistor (load).
