Resistance and Ohm's Law

Apply the same voltage to different sections: the currents may differ. If we vary the voltage across one section, will the current always change in the same proportion?

The field directs electrons in a metal, but their motion is not free acceleration through empty space. They interact with the material and transfer energy to it. Different sections therefore carry different currents at the same voltage. Resistance R describes this property and is measured in ohms. At the same voltage, greater resistance means less current. Charge does not disappear inside a resistor.

Take a resistor, vary its voltage and measure the current. Try to keep temperature and all other conditions unchanged. If doubling U doubles I, halving U halves I, and U/I stays constant throughout the tested range, the section is called ohmic. This is an experimental result, not a rule imposed on every device. One measured current then lets us predict the others under the same conditions.

Call the constant ratio U/I resistance R and write Ohm's law: I = U/R. With U in V and R in ohms, I is in A. The law applies to an ohmic section with no source inside it, at practically unchanged temperature and state. At U = 0 its current is zero; as R increases, the same voltage source produces less current. At R = 0 division cannot be used: the idealisation no longer describes the real circuit.

How is the resistance unit defined? If 1 V produces 1 A, the resistance is 1 ohm: 1 Ом = 1 В/А. From a measured nonzero current and voltage, calculate R = U/I. But one measurement gives the ratio only under those conditions. To claim R is constant, test several voltages. Thus the definition of resistance alone does not prove Ohm's law over the whole range.

Plot the results with U on the horizontal axis and I on the vertical axis. An ohmic section gives a straight line through the origin: each voltage change produces a proportional current change. A steeper line means more current at the same U, hence a smaller R. If the axes are swapped to plot U(I), the conclusion about steepness changes. Always read the axis labels first.

Current heats a lamp filament and changes its resistance. As voltage increases, current therefore usually does not keep rising in the same proportion: the graph curves. Can we restore the rule simply by keeping any device at constant temperature? No: a diode, for example, has a nonlinear relationship even without such heating. Constant temperature matters, but ohmic behaviour must still be established experimentally for each section.

Check two different comparisons. For the same ohmic resistor with unchanged R, doubling U doubles I. Now keep U unchanged and replace the resistor with one of twice the R: current halves. Doubling both U and R leaves the current unchanged. Saying 'resistance increased, so current decreased' is incomplete unless we say what happened to voltage and temperature.

A 6 ohm resistor has 12 V applied across it; its temperature changes very little. Find the current.

Treat this resistor as ohmic under the stated conditions, with nearly constant temperature. Current is related to voltage and resistance by I = U/R.

I = 12/6 = 2 А. Check backwards: 2 А·6 Ом = 12 В, recovering the units and original voltage.

Replacing it with 12 ohms while keeping 12 V would give 1 A. Twice the resistance lets half as much charge pass per second.

The current is 2 A. In steady operation, 2 C passes through the resistor's cross-section each second.

Ohm's law is the experimentally established proportionality of I and U for a particular ohmic section under unchanged conditions.

The ratio U/I is measured in ohms; a single measurement does not prove that R is constant.

When predicting a current change, state what stays fixed: voltage, resistance and the section's physical state.

Next we will find how resistance depends on a wire's material, length and thickness.