Maxwell in differential form

The four Maxwell equations written so far are in integral form: they speak of fluxes through surfaces and circulations along closed curves. There is an equivalent, more compact form that rewrites them at every point in space, without integrals:

$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla\cdot\vec{B} = 0$

$\nabla\wedge\vec{E} = -\frac{\partial\vec{B}}{\partial t}, \qquad \nabla\wedge\vec{B} = \mu_0\,\vec{j} + \mu_0\varepsilon_0\,\frac{\partial\vec{E}}{\partial t}$

The symbol $\nabla$ (nabla) is a “vector of partial derivatives” $\left(\partial/\partial x,\;\partial/\partial y,\;\partial/\partial z\right)$:

• $\nabla\cdot\vec{F}$ (divergence) measures how much a field “originates from that point”, i.e. its point sources;
• $\nabla\wedge\vec{F}$ (curl) measures how much a field “rotates around that point”, i.e. its local vortices.

Historical context

The differential form is the one in which Maxwell wrote his 1865 treatise (using Hamilton’s quaternion notation); the modern vector form is due to Oliver Heaviside, in the 1880s. For numerical applications — every simulation of antennas, microwaves, gravitational waves — it is the form used in practice. For sixth-form purposes it is enough to know that it exists and is fully equivalent to the integral form (Mazzoldi 2008).

Links

Topics: Electromagnetic waves Concepts: Maxwell’s equations

Related exercises: Problem — Speed of light from vacuum constants · Problem — E-B symmetry and asymmetries · Problem — EM waves without a medium