# 9.4 Ohm's law  (Page 2/7)

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In this experiment, the voltage applied across the resistor varies from −10.00 to +10.00 V, by increments of 1.00 V. The current through the resistor and the voltage across the resistor are measured. A plot is made of the voltage versus the current, and the result is approximately linear. The slope of the line is the resistance, or the voltage divided by the current. This result is known as Ohm’s law    :

$V=IR,$

where V is the voltage measured in volts across the object in question, I is the current measured through the object in amps, and R is the resistance in units of ohms. As stated previously, any device that shows a linear relationship between the voltage and the current is known as an ohmic device. A resistor is therefore an ohmic device.

## Measuring resistance

A carbon resistor at room temperature $\left(20\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\right)$ is attached to a 9.00-V battery and the current measured through the resistor is 3.00 mA. (a) What is the resistance of the resistor measured in ohms? (b) If the temperature of the resistor is increased to $60\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ by heating the resistor, what is the current through the resistor?

## Strategy

(a) The resistance can be found using Ohm’s law. Ohm’s law states that $V=IR$ , so the resistance can be found using $R=V\text{/}I$ .

(b) First, the resistance is temperature dependent so the new resistance after the resistor has been heated can be found using $R={R}_{0}\left(1+\alpha \text{Δ}T\right)$ . The current can be found using Ohm’s law in the form $I=V\text{/}R$ .

## Solution

1. Using Ohm’s law and solving for the resistance yields the resistance at room temperature:
$R=\frac{V}{I}=\frac{9.00\phantom{\rule{0.2em}{0ex}}\text{V}}{3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{A}}=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}=3.00\phantom{\rule{0.2em}{0ex}}\text{k}\phantom{\rule{0.2em}{0ex}}\text{Ω}.$
2. The resistance at $60\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ can be found using $R={R}_{0}\left(1+\alpha \text{Δ}T\right)$ where the temperature coefficient for carbon is $\alpha =-0.0005$ . $R={R}_{0}\left(1+\alpha \text{Δ}T\right)=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\left(1-0.0005\left(60\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}-20\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}\right)\right)=2.94\phantom{\rule{0.2em}{0ex}}\text{k}\phantom{\rule{0.2em}{0ex}}\text{Ω}$ .
The current through the heated resistor is
$I=\frac{V}{R}=\frac{9.00\phantom{\rule{0.2em}{0ex}}\text{V}}{2.94\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}}=3.06\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\text{A}=3.06\phantom{\rule{0.2em}{0ex}}\text{mA}.$

## Significance

A change in temperature of $40\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ resulted in a 2.00% change in current. This may not seem like a very great change, but changing electrical characteristics can have a strong effect on the circuits. For this reason, many electronic appliances, such as computers, contain fans to remove the heat dissipated by components in the electric circuits.

Check Your Understanding The voltage supplied to your house varies as $V\left(t\right)={V}_{\text{max}}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\left(2\pi ft\right)$ . If a resistor is connected across this voltage, will Ohm’s law $V=IR$ still be valid?

Yes, Ohm’s law is still valid. At every point in time the current is equal to $I\left(t\right)=V\left(t\right)\text{/}R$ , so the current is also a function of time, $I\left(t\right)=\frac{{V}_{\text{max}}}{R}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\left(2\pi ft\right)$ .

See how the equation form of Ohm’s law relates to a simple circuit. Adjust the voltage and resistance, and see the current change according to Ohm’s law. The sizes of the symbols in the equation change to match the circuit diagram.

Nonohmic devices do not exhibit a linear relationship between the voltage and the current. One such device is the semiconducting circuit element known as a diode. A diode    is a circuit device that allows current flow in only one direction. A diagram of a simple circuit consisting of a battery, a diode, and a resistor is shown in [link] . Although we do not cover the theory of the diode in this section, the diode can be tested to see if it is an ohmic or a nonohmic device.

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