Ohm’s law expresses a fundamental idea in electromagnetism and materials physics: that the current density J responds linearly to the applied electromagnetic force acting on the charges.
This relation defines the concept of electrical conductivity. At its core, Ohm’s law tells us that:
J=σFem,
where:
- J is the electric current density (A/m²),
- σ is the electrical conductivity (S/m),
- Fem is the electromagnetic force per unit charge (V/m), which may include both electric and magnetic components.
This is a constitutive relation: it describes how a material or medium responds to electromagnetic fields. The force per unit charge is given by the Lorentz force:
Fem=E+v×B.
Hence, in its general form:
J=σ(E+v×B).
This expression is particularly relevant in conducting fluids like plasmas, where the bulk velocity v is nonzero.
In many ordinary conductors (like metals), the motion of the conducting medium itself is negligible — that is, v≈0. In that case, we recover the familiar textbook version of Ohm’s law:
J=σE
This relation holds when:
- The material is isotropic (same in all directions),
- The temperature and conductivity are constant,
- There are no magnetic effects or significant inertial contributions.
This is the form most often encountered in introductory physics and electrical engineering contexts.
To understand the origin of Ohm’s law from first principles, consider the classical Drude model of electrical conduction.
We model conduction electrons as classical particles that:
- Accelerate under an electric field E,
- Scatter randomly off atoms with an average time between collisions τ.
Under these conditions, the equation of motion for an electron is
mdtdv=−eE−τmv.
- The first term is the electric force.
- The second term represents the resistive drag from collisions (a friction-like term).
In the steady state (dv/dt=0), the average drift velocity is:
vdrift=−meτE
With n conduction electrons per unit volume:
J=−nevdrift=(mne2τ)E
This gives us the microscopic expression for conductivity:
σ=mne2τ
Thus, Ohm's law emerges naturally from basic classical mechanics and statistical assumptions about electron motion.
¶ Generalized Ohm’s Law (Plasmas and MHD)
In plasmas or moving conductors, we must take into account the full Lorentz force acting on the charges. This leads to the generalized Ohm’s law:
J=σ(E+v×B)
- This form is valid in the rest frame of the conducting fluid.
- It accounts for motional electromotive forces (EMF), which arise from movement through a magnetic field.
- It is central to plasma physics and forms the foundation of magnetohydrodynamics (MHD).
In many astrophysical and space plasma contexts, the plasma is assumed to be an excellent conductor: σ→∞, or equivalently η=1/σ→0 (zero resistivity).
In this limit, Ohm’s law becomes:
E+v×B=0
This implies the electric field vanishes in the fluid rest frame, and leads to the important concept of magnetic field lines being frozen into the plasma, meaning they are "dragged" along with the fluid motion.
In more general cases, such as collisionless plasmas or magnetic reconnection regions, Ohm’s law includes additional terms:
E+v×B=ηJ+ne1J×B−ne1∇pe+…
- The Hall term (J×B)/(ne) becomes important when ion and electron motions decouple.
- The electron pressure gradient term accounts for thermal effects.
- Electron inertia terms may be included for high-frequency or small-scale dynamics.
These extensions will be explored in more detail in the MHD section of the wiki.
Ohm’s law encapsulates how electric fields drive currents — whether in simple metals or in the swirling magnetized plasmas of astrophysical environments. While its classical form J=σE is widely used, its true power lies in its generalizations, which describe how fields and fluids interact in dynamic, high-energy systems like stars, jets, and accretion disks.
- Griffiths, D. J., Introduction to Electrodynamics, 4th ed., Pearson (2013), Ch. 7
- Jackson, J. D., Classical Electrodynamics, 3rd ed., Wiley (1998), Sec. 5.12
- Kulsrud, R. M., Plasma Physics for Astrophysics, Princeton University Press (2005)
- Goedbloed, J. P., & Poedts, S., Principles of Magnetohydrodynamics, Cambridge (2004)