## Table of contents

**Ohm’s law** describes the correlation of electrical quantities (resistance, current, voltage) as they vary. Resistance (R) means the obstacle that meets the current in its path, the higher it is the more difficult it will be for the current to cross it; the unit of measure is represented by the greek letter omega (Ω).

Current (I) means the intensity of electric charges that travel through a conductor. Voltage (V) is the potential difference between one point and another, expressed in volts. Clearly the higher the obstacle is, the lower will be the intensity of current flowing through it, it is clear that current is inversely proportional to resistance. For the voltage instead the higher it is, the greater is its attraction force that generates to move the charges, consequently the same resistive value will be directly proportional to the current.

The name is due to the German physicist Georg Ohm, who, in a treatise published in 1827, described the measurement of current and potential difference through simple circuits with wires of different lengths, although the original formulation is more complex than the current form.

Ohm’s law can be used to validate the static values of a circuit’s components, current levels, supply voltages, and voltage drops. For example, if a measuring instrument detects a higher-than-normal current measurement, it could mean that the resistance has decreased or the voltage has increased, resulting in a high-voltage situation. This could indicate a power supply or circuit problem.

In direct current (DC) circuits, a lower-than-normal current measurement could indicate that the voltage has decreased or that the circuit resistance has increased. Possible causes of increased resistance are faulty or loose connections, corrosion, and/or damaged components.

Loads within a circuit draw electrical current. Loads can be a variety of components: small electrical devices, computers, appliances, or a powerful motor. Most of these components (loads) have a nameplate or an informational sticker. These nameplates contain safety certification and multiple reference values.

Technicians consult the nameplates on the components for voltage and current reference values. If, during testing, technicians find that their digital multimeters or clamp meters do not register the usual values, they can use Ohm’s law to detect which part of the circuit is working abnormally and then determine where the problem may lie.

## First Ohm’s law

First Ohm’s lawstates that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:\[I=\dfrac{V}{R}\]

where \(I\) is the current through the conductor in units of amperes, \(V\) is the voltage measured across the conductor in units of volts, and \(R\) is the resistance of the conductor in units of ohms (Ω).

Ohm’s first law describes a simple relationship between the three fundamental quantities concerning a resistor. It is an empirical law, that is identified by experiments and not demonstrated by theory. If we try to measure the voltage for different values of current flowing through the same conductor, we find that voltage and current are related by a linear relationship. If we draw a graph of voltage (on y-axis) versus current (on x-axis) we obtain a straight line through the origin. In particular the passage of the line through the origin implies that to a zero current corresponds a zero voltage. The resistance is instead the coefficient of proportionality between voltage and current, so it corresponds to the angular coefficient of the line and expresses the slope of the line with respect to the horizontal axis.

It is important to specify that not all conductors respect Ohm’s first law, and those that satisfy it are called ohmic conductors. For some conductors the linear dependence between potential difference and current intensity is not valid, and between these quantities there is another kind of dependence (as it happens for example for LED lights). Normally, however, when dealing with normal circuits, the conductors considered always satisfy Ohm’s first law: for this reason in the lessons on Electrodynamics we will always implicitly assume to work with ohmic conductors, unless otherwise specified.

## Second Ohm’s law

The first law tells us how resistance is related to potential difference and current intensity, but it doesn’t tell us on which factors it depends. More precisely we have seen that electrical resistance depends on the conductor (so the material that constitutes it, its size and shape) and some external conditions (like temperature).

In this regard we define an additional quantity, the electrical resistivity, which is indicated by the Greek letter ρ and expresses the tendency of the material to oppose the passage of current. This eliminates the dependence on other characteristics of the conductor, while preserving the dependence on external factors.

\[R=\rho\dfrac{l}{S}\]

Where by (l) we denote the length of the resistor and by (S) the cross-sectional area. Ohm’s second law states that the resistance of a conductor is directly proportional to its length and inversely proportional to the cross sectional area, and also that it depends on the material of which the conductor is made.

With cross section we usually mean the cross section of the conductor perpendicular to the motion of electric charges. Let’s consider for example a prism with square base and imagine it placed vertically; then imagine to place it horizontally and make it crossed by a current also horizontally: in this case the cross section will be the square base of the prism. In this situation the length is given by the height of the prism.

## Limits of Ohm’s Law

Ohm’s law is generalized to most materials. It is a less general law than Maxwell’s equations, and in some materials it does not hold. In metals the law has a universal character, while in insulators it applies only to weak local electric fields. In fact in insulators the drift speed of electrons can reach very high values and in this case there is a dielectric breakdown.

Ohm’s law has been observed at different scales. At the beginning of the twentieth century it was believed that Ohm’s law should lose its validity for dimensions comparable to atomic spacing, but in 2012 it was experimentally demonstrated that a strip of silicon four atoms wide and one atom thick still complies with Ohm’s law.