Other versions of Ohm's law

Other versions of Ohm's law

Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law:^[23]

where "E" is the electric field vector with units of volts per meter (analogous to "V" of Ohm's law which has units of volts), "J" is the current density vector with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "ρ" (Greek "rho") is theresistivity with units of ohm·meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written^[24] asJ = σE where "σ" is the conductivity which is the reciprocal of ρ.

Current flowing through a uniform cylindrical conductor (such as a round wire) with a uniform field applied.

The potential difference between two points is defined as:^[25]

with  the element of path along the integration of electric field vector E. If the appliedE field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

Since the E field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density J will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:^[26]

Substituting the above 2 results (for E and J respectively) into the continuum form shown at the beginning of this section:

The electrical resistance of a uniform conductor is given in terms of resistivity by:^[26]

where l is the length of the conductor in SI units of meters, a is the cross-sectional area (for a round wire a = πr² if r is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.

After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form:

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,^[27] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.

The more complex generalized forms of Ohm's law are important to condensed matter physics, which studies the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of transport coefficients.

Magnetic effects

If an external B-field is present and the conductor is not at rest but moving at velocity v, then an extra term must be added to account for the current induced by the Lorentz force on the charge carriers.

In the rest frame of the moving conductor this term drops out because v= 0. There is no contradiction because the electric field in the rest frame differs from the E-field in the lab frame: E ' = E + vxB. Electric and magnetic fields are relative, see Lorentz transform.

If the current J is alternating because the applied voltage or E-field varies in time, then reactance must be added to resistance to account for self-inductance, see electrical impedance. The reactance may be strong if the frequency is high or the conductor is coiled.

See Hall effect for some other implication of a magnetic field.