Ohm's Law

If an electric conductor is connected to a voltage source, e.g. a battery, a current can be measured. Experimentally it turned out that this current is directly proportional to the applied voltage. This is usually summarized with the following formula. This equation is called Ohm's law. The inverse proportional constant $R$ between $I$ and $U$ is called resistant and has the unit $\Omega$ which is pronounced Ohm, named after Georg Simon Ohm. We want to "derive" this formula by making it plausible by taking a look at the microscopic charge motion inside a wire where the moving always collides with positively charged atomic cores. Due to these collisions, they travel with a constant drift speed which is equal to the acceleration due to the applied electric field times the time $\tau$ between two collisions: $$\vec{v} = \vec{a}\tau = \frac{\vec{F}}{m}\tau$$ We can now write the charge density as $$\vec{j} = \varrho \vec{v} = nq\frac{\vec{F}}{m}\tau$$ In the last step, $\varrho$ was replaced with $nq$ where $n$ is the number per unit volume and $q$ is the total charge of all electrons. Now we can insert the formula $\vec{F} = q\vec{E}$ which leads to $$\vec{j} = \frac{nq^2\tau}{m}\vec{E}$$ The term $nq^2\tau/m$ is a material constant called conductivity: $$\boxed{\sigma_\mathrm{el} = \frac{nq^2\tau}{m}}$$ In addition to that, we can also define resistivity as the inverse conductivity $$\boxed{\varrho_\mathrm{el} = \frac{1}{\sigma_\mathrm{el}}}$$ If $E$ is replaced with $U/l$ and $j$ with $I/A$ it follows $$I = \frac{1}{\varrho_\mathrm{el}}\frac{A}{l} U$$ For every wire with the length $l$ and the cross-section $A$, the product of constants can be combined to the resistance $R$ which is then given as $$\boxed{R = \varrho_\mathrm{el}\frac{l}{A}}$$ The unit of the resistivity can therefore is usually given as $\Omega\,\mathrm{mm}$ This formula can easily be memorized. Since a resistor consists basically of a simple short wire, its resistance increases with the length of the wire, because the electrons lose a larger amount of energy to the atomic cores. On the other hand, a larger diameter increases the number of electrons which decreases the resistance. Now we can rewrite the relation between $I$ and $U$ according to $$I = \frac{U}{R}$$ This is the previously stated very famous and important Ohm's law for conductors. Almost every material obeys that law. Exceptions from that law can mainly be found doped in doped semiconductors like diodes or transistors.