In an electrical conductor, electrons are constantly in motion due to thermal and quantum effects. Their speed can reach up to about 10% of the speed of light. However, since this motion occurs equally in all directions, the electrons remain on average at the same location within the conductor. When an electric voltage, and thus an electric field, is applied to the conductor, the electrons experience a force that accelerates them. Collisions with the atomic nuclei in the conductor lead to a constant velocity known as the drift velocity. This velocity must be proportional to the electric field and therefore to the voltage. Theoretically, this drift velocity is in the range of a few centimeters per second. Since electric current is proportional to this velocity, there is a linear relationship between current and the applied voltage.
Experimentally, this law can be confirmed with a simple circuit: when a voltage is applied to a wire, the current increases proportionally with the voltage. According to Kirchhoff’s loop rule, the voltage across the resistor must equal the applied voltage $U$, assuming the internal resistance of the ammeter is negligible. Therefore, the proportionality
$$
U \propto I
$$
holds. The proportionality constant is called the Ohmic resistance and is usually denoted by $R$:
$$
\Delta U = R \Delta I
$$
Thus, Ohm’s law — also known as the URI rule — can be expressed in one of its three forms as:
$$
R = \frac{U}{I}
$$
The unit of electrical resistance is the ohm ($\Omega$).
Example: A 4.7 k$\Omega$ resistor is connected to a voltage source of 1.5 V. What current flows through the circuit? Answer: 0.26 mA.
A useful analogy can be made with the flow of fluids in pipes: if we replace the volume flow in Hagen–Poiseuille’s law with electric current and the pressure drop with voltage, we obtain Ohm’s law directly:
$$
I = \frac{U}{R}
$$
Here, the reciprocal of the constant is summarized as $R$. Just as the volume flow in a pipe system must remain constant, the current in a closed electric circuit must also remain constant. This follows from the fact that charge cannot be created, destroyed, or take a path other than through the conductor. Therefore, if several resistors are present in a circuit, the same current $I$ flows through all of them, even though the resistors may have different values. At each resistor, a voltage drop occurs, which can be calculated using Ohm’s law:
$$
U = R I
$$
This principle is applied in voltage dividers, which in their simplest form consist of multiple resistors connected in series to a voltage source. Across each resistor, a different voltage can then be measured and tapped.