20.2 Ohm’s Law: Resistance and Simple Circuits

Ohm's Law

The current that flows through most substances is directly proportional to the voltage $V$ applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied:

This important relationship is known as Ohm's law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn't always occur.

Resistance and Simple Circuits

If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance $R$. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or

Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives

This relationship is also called Ohm's law. Ohm's law in this form really defines resistance for certain materials. Ohm's law (like Hooke's law) is not universally valid. The many substances for which Ohm's law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance $R$ that is independent of voltage $V$ and current $I$. An object that has simple resistance is called a resistor, even if its resistance is small. The unit for resistance is an ohm and is given the symbol $\Omega$ (upper case Greek omega). Rearranging $I=\text{V/R}$ gives $R=\text{V/I}$, and so the units of resistance are 1 ohm = 1 volt per ampere:

Figure 20.8 shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in $R$.

Figure 20.8 A simple electric circuit in which a closed path for current to flow is supplied by conductors (usually metal wires) connecting a load to the terminals of a battery, represented by the red parallel lines. The zigzag symbol represents the single resistor and includes any resistance in the connections to the voltage source.

Making Connections: Real World Connections

Ohm's law ($V=IR$) is a fundamental relationship that could be presented by a linear function with the slope of the line being the resistance. The resistance represents the voltage that needs to be applied to the resistor to create a current of 1 A through the circuit. The graph (in the figure below) shows this representation for two simple circuits with resistors that have different resistances and thus different slopes.

Figure 20.9 The figure illustrates the relationship between current and voltage for two different resistors. The slope of the graph represents the resistance value, which is 2Ω and 4Ω for the two lines shown.

Making Connections: Real World Connections

The materials which follow Ohm's law by having a linear relationship between voltage and current are known as ohmic materials. On the other hand, some materials exhibit a nonlinear voltage-current relationship and hence are known as non-ohmic materials. The figure below shows current voltage relationships for the two types of materials.

Figure 20.10 The relationship between voltage and current for ohmic and non-ohmic materials are shown.

Clearly the resistance of an ohmic material (shown in (a)) remains constant and can be calculated by finding the slope of the graph but that is not true for a non-ohmic material (shown in (b)).

Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of ${\text{10}}^{\text{12}}\Omega$ or more. A dry person may have a hand-to-foot resistance of ${\text{10}}^5\Omega$, whereas the resistance of the human heart is about ${\text{10}}^3\Omega$. A meter-long piece of large-diameter copper wire may have a resistance of ${\text{10}}^{-5}\Omega$, and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity.

Additional insight is gained by solving $I=\text{V/R}$ for $V,$ yielding

This expression for $V$ can be interpreted as the voltage drop across a resistor produced by the current $I$. The phrase $\text{IR}$ drop is often used for this voltage. For instance, the headlight in Example 20.4 has an $\text{IR}$ drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since $\text{PE}=q\mathrm{\Delta }V$, and the same $q$ flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See Figure 20.11.)

Figure 20.11 The voltage drop across a resistor in a simple circuit equals the voltage output of the battery.