5.4.5. Ohm's Law

Table of Contents

Linear relation between voltage and current

Ohm's law describes a simple and very important behavior of many electrical conductors. It states that, for an ohmic conductor under fixed physical conditions such as temperature, the current through the conductor is directly proportional to the potential difference across it.

If a conductor obeys this rule, then doubling the voltage doubles the current, tripling the voltage triples the current, and so on. The constant of proportionality is called the resistance.

The mathematical form is

$$
V = IR
$$

where $V$ is the potential difference, $I$ is the current, and $R$ is the resistance.

For an ohmic conductor at constant temperature,
$$
V = IR
$$
This means
$$
I = \frac{V}{R}
\qquad \text{and} \qquad
R = \frac{V}{I}
$$
The law applies only when the resistance remains constant.

Meaning of the quantities

Voltage measures how much electric potential difference pushes charge through a conductor. Current tells how much charge passes per unit time. Resistance tells how strongly the conductor opposes the flow of current.

In Ohm's law, resistance is the slope factor connecting voltage and current. A large resistance means a small current for the same voltage. A small resistance means a larger current for the same voltage.

The unit relation is

$$
1 \ \Omega = 1 \ \frac{\text{V}}{\text{A}}
$$

where $\Omega$ is the ohm.

Ohmic behavior

A material or device is called ohmic if the ratio $V/I$ stays constant. In that case, a graph of voltage against current is a straight line through the origin.

This does not mean every electrical device obeys Ohm's law. Some devices, such as filament lamps, diodes, and many semiconductors, have resistance that changes with voltage, current, or temperature. For such devices, the current is not proportional to voltage over the whole range.

Ohm's law is not a universal law for every circuit element.
It is valid for elements whose resistance is constant under the given conditions.

Graphical interpretation

For an ohmic resistor, the $V$ versus $I$ graph is linear:

$$
V \propto I
$$

The slope of a $V$ versus $I$ graph is

$$
\frac{V}{I} = R
$$

So a steeper line means a larger resistance.

If instead you draw $I$ versus $V$, then the slope is

$$
\frac{I}{V} = \frac{1}{R}
$$

which is called the conductance.

Rendering scientific sketch…

Voltage-current graph for an ohmic resistor

Rearranging the law

Ohm's law can be used in three equivalent ways depending on what quantity is unknown.

If voltage is unknown,

$$
V = IR
$$

If current is unknown,

$$
I = \frac{V}{R}
$$

If resistance is unknown,

$$
R = \frac{V}{I}
$$

These forms are all algebraically the same. The correct form depends only on what you want to calculate.

Simple numerical examples

Suppose a resistor of $4 \ \Omega$ is connected across a voltage of $12 \ \text{V}$. The current is

$$
I = \frac{V}{R} = \frac{12}{4} = 3 \ \text{A}
$$

Now suppose a current of $0.50 \ \text{A}$ flows through a resistor of $20 \ \Omega$. The voltage across it is

$$
V = IR = (0.50)(20) = 10 \ \text{V}
$$

If a device has a voltage of $9 \ \text{V}$ across it and current $3 \ \text{A}$ through it, then its resistance is

$$
R = \frac{V}{I} = \frac{9}{3} = 3 \ \Omega
$$

Table of variable relationships

Quantity	Symbol	Unit	From Ohm's law
Voltage	$V$	volt, $\text{V}$	$V = IR$
Current	$I$	ampere, $\text{A}$	$I = \frac{V}{R}$
Resistance	$R$	ohm, $\Omega$	$R = \frac{V}{I}$

Physical conditions matter

The statement of Ohm's law usually assumes that the conductor's physical state does not change. Temperature is especially important. Many conductors heat up when current flows, and heating can change the resistance. If resistance changes, the simple proportional relation may no longer hold exactly.

For this reason, when we say a resistor obeys Ohm's law, we usually mean over a certain range and at roughly constant temperature.

When using Ohm's law, assume constant resistance only if the material conditions are unchanged, especially temperature.

Microscopic picture

Inside a conductor, moving charges are pushed by the electric field created by the voltage. The material resists this motion because charges collide with atoms and imperfections in the material. For many metals, this produces a nearly proportional relation between current and voltage, which is why Ohm's law works well.

This chapter does not require the full microscopic theory, but it helps to remember that resistance comes from the material's opposition to charge flow.

Distinguishing law from definition

It is easy to confuse the formula $R = V/I$ with a definition that always applies. For any device, you can calculate the ratio $V/I$ at a particular operating point. But only if that ratio stays constant as $V$ and $I$ change do we say the device obeys Ohm's law.

So the formula alone is not enough. The key idea is proportionality.

Practical use

Ohm's law is one of the most frequently used tools in circuit calculations. When two of the three quantities are known, the third can be found immediately. It is the starting point for analyzing resistors in simple and complex circuits.

In later circuit work, Ohm's law is applied repeatedly to different parts of a circuit, but the core idea remains the same, voltage, current, and resistance are linked by a linear rule for ohmic elements.

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