Electricity and magnetism

Ohm's law for a circuit segment

Current is proportional to voltage and inversely proportional to resistance: I = U/R.

Ohm's law is the empirical rule describing the relationship between the voltage applied across a circuit segment and the current flowing through it, via resistance $R$ .

I = \frac{U}{R}

$U$ is the voltage (volts, V) across the segment, $I$ is the current (amperes, A), and $R$ is resistance (ohms, Ω) — a quantity that characterizes how much a material and wire shape resist current flow.

Definition: Resistance

The resistance of a uniform cylindrical wire depends on the material's resistivity $\rho$ , its length $l$ , and cross-section $S$ : $R = \rho l/S$ . A longer, thinner wire has more resistance — much like a long, narrow pipe resists flow more.

Watch electrons (blue dots) drift through the wire at a speed proportional to I. Drag the U, R sliders to see I = U/R change live on the graph.

Series and parallel circuits

When two resistors $R_1, R_2$ are connected in series, the same current $I$ flows through both, while the voltage splits in proportion to resistance:

R_{\text{td}} = R_1 + R_2, \qquad U_1 = IR_1,\ U_2 = IR_2

When connected in parallel, both branches share the same voltage $U$ , while the current splits inversely with resistance:

\frac{1}{R_{\text{td}}} = \frac{1}{R_1} + \frac{1}{R_2}, \qquad I_1 = \frac{U}{R_1},\ I_2 = \frac{U}{R_2}

Example: Parallel circuit

Two resistors $R_1 = 6\ \Omega$ and $R_2 = 12\ \Omega$ are connected in parallel across a $U = 12$ V source. Find the current in each branch and the total.

Solution

$I_1 = U/R_1 = 12/6 = 2$ A, $I_2 = U/R_2 = 12/12 = 1$ A. Total current $I = I_1 + I_2 = 3$ A, consistent with $R_{\text{eq}} = 4\ \Omega$ and $I = U/R_{\text{eq}} = 12/4 = 3$ A.

Ohm’s law is an empirical relation in the ohmic operating regime: at stable temperature and material conditions, doubling voltage doubles current. A graph of $I$ against $U$ is a straight line through the origin with slope $1/R$ . A lamp filament is not always linear because it heats as current flows; distinguish resistance at a particular state from an ohmic component whose $R$ is nearly constant.

Resistance can be determined by measuring the voltage across a resistor and the current through it. If $U=9.0$ V and $I=0.30$ A, then $R=30\,\Omega$ . When connecting a load, use consistent SI units and check its dissipated power, $P=UI=2.7$ W, as a reasonableness test for the measured current.

Quick check

Keeping voltage fixed, if the circuit's resistance triples, the current will:

Two equal resistors in parallel give an equivalent resistance equal to:

References

Georg Simon Ohm (1827). Die galvanische Kette, mathematisch bearbeitet
Young, Freedman (2019). University Physics