Ohm's Law

A conductor is a medium that contains free electric charges (usually electrons) that acquire a net drift velocity in the presence of an applied electric field, giving rise to an electric current flowing in the same direction as the field. The well-known relationship between the current and the voltage in a typical conductor is given by Ohm's law:

$$V = I\, R,$$ (2.113)

where $V$ is the voltage drop across a conductor of electrical resistance $R$ through which a current $I$ flows. Incidentally, the unit of electric current is the ampere (or amp) (A), which is equivalent to a coulomb per second. Furthermore, the unit of electrical resistance is the ohm ($\Omega$), which is equivalent to a volt per ampere.

Let us generalize Ohm's law so that it is expressed in terms of the electric field, ${\bf E}$, and current density, ${\bf j}$, at a given point inside the conductor, rather than the global quantities $V$ and $I$. Here, the magnitude of the current density vector, ${\bf j}$, measures the amount of current flowing per unit time per unit cross-sectional area, whereas the direction of the vector specifies the direction of the current flow. Consider a length $l$ of a conductor of uniform cross-sectional area $A$ through which a net electric current $I$ flows. In general, we expect the electrical resistance of the conductor to be proportional to its length, $l$, and inversely proportional to its cross-sectional area, $A$ (i.e., we expect it to be harder to push an electrical current down a long rather than a short wire, and easier to push an electrical current down a wide rather than a narrow conducting channel.) Thus, we can write

$$R = \eta\, \frac{l}{A}.$$ (2.114)

Here, the constant $\eta$ is called the resistivity of the conducting medium, and is measured in units of ohm-meters. Hence, Ohm's law becomes

$$V = \eta\, \frac{l}{A}\, I.$$ (2.115)

However, $I/A = j_z$ (supposing that the conductor is aligned along the $z$-axis) and $V/l = E_z$ [see Equation (2.17)], so the previous equation reduces to

$$E_z = \eta \,j_z.$$ (2.116)

Because there is nothing special about the $z$-axis (in an isotropic conducting medium), the previous formula immediately generalizes to

$${\bf E} = \eta \,{\bf j}.$$ (2.117)

This is the most fundamental form of Ohm's law.

It is fairly easy to account for the previous equation at the microscopic level. Consider a metal that has $n_e$ free electrons per unit volume. Of course, the metal also has a fixed lattice of metal ions whose charge per unit volume is equal and opposite to that of the free electrons, rendering the medium electrically neutral. In the presence of an electric field ${\bf E}$, a given free electron is subject to an electrical force ${\bf f} = -e\,{\bf E}$ [see Equation (2.10)], and therefore accelerates (from rest at $t=0$) such that its drift velocity is written ${\bf v} = -(e/m_e)\,t\,{\bf E}$, where $-e$ is the electron charge, and $m_e$ the electron mass. Suppose that, on average, a drifting electron collides with a metal ion once every $\tau$ seconds. Given that a metal ion is much more massive than an electron, we expect a free electron to lose all of the momentum it had previously acquired from the electric field during such a collision. It follows that the mean drift velocity of the free electrons is $\langle{\bf v}\rangle = -(e\,\tau/2\,m_e)\,{\bf E}$. Hence, the mean current density is

$${\bf j} = -n_e\,e\,\langle{\bf v}\rangle= \frac{n_e\,e^2\,\tau}{2\,m_e}\,{\bf E}.$$ (2.118)

Thus, we can account for Equation (2.117), as long as the resistivity takes the form

$$\eta = \frac{2\,m_e}{n_e\,e^2\,\tau}.$$ (2.119)

We conclude that the resistivity of a typical conducting medium is determined by the number density of free electrons, as well as the mean collision rate of these electrons with the fixed ions.

A free charge $q$ that moves through a voltage drop $V$ acquires an energy $q\,V$ from the electric field. (See Section 2.1.5.) In a conducting medium, this energy is dissipated as heat (the conversion to heat takes place each time a free charge collides with a fixed ion). This particular type of heating is called ohmic heating or Joule heating. Suppose that $N$ charges per unit time pass through a conductor. The current flowing through the conductor is obviously $I= N\,q$. The total energy gained by the charges, which appears as heat inside the conductor, is

$$P = N\,q\,V = I\,V$$ (2.120)

per unit time. Thus, the heating power is

$$P = I\,V = I^2 \,R = \frac{V^2}{R}.$$ (2.121)

Equations (2.120) and (2.121) generalize to

$$P = {\bf j} \cdot {\bf E} = \eta \,j^{\,2},$$ (2.122)

where $P$ is now the power dissipated per unit volume inside the conducting medium.