Visualising Maxwell's Equations

Maxwell's equations are a kind of pons asinorum of physics. An essential bridge to be crossed in electromagnetism - yet as a system of equations - apparently impenetrable. First obstacle: we have those weird operators: div, grad and curl; second obstacle: the vector differential operator del (which when applied directly is the operator grad)is commingled with the dot and cross product operations to somehow produce div and curl.

I tend to understand things visually and geometrically: I want to think that the equations describe the time and space behaviour of some entity or entities which I can visualise as a whole. It is said that Maxwell was influenced by mechanical analogies, imagining electric and magnetic fields as if they were filled with tiny gears, wheels, and cogs in space. This approach enabled him to think in terms of physical interactions and visualize the fields as interconnected, dynamic entities rather than as the purely abstract concepts denoted by the equations without context.

I asked ChatGPT to take me through Maxwell's equations and highlight their geometric nature, starting with the operators.

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Divergence (div or 'del dot') - ∇ ·

∇ ⋅ F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

The divergence of a field measures how much the field spreads out from or converges into a point.

In Maxwell's equations:

Gauss's law for electricity: ∇ · E = ρ/ε₀. This shows that the divergence of the electric field E is proportional to the local charge density ρ, meaning electric field lines radiate out from positive charges and converge into negative charges.

Gauss's law for magnetism: ∇ · B = 0. The divergence of the magnetic field B is always zero, indicating that magnetic field lines form closed loops, with no magnetic monopoles acting as sources or sinks.

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Gradient (grad) - ∇

∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)

The gradient of a scalar field represents the direction and rate of its steepest increase. In the case of the electric field:

The electric field E can be derived from the electric potential φ as E = -∇φ. This means that the electric field points in the direction of the steepest decrease in electric potential.

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Curl (rot or 'del cross') - ∇ ×

The curl of a vector field measures its tendency to circulate around a point, describing the "rotational" aspect of the field. In determinant form:

 

∇ × F =

i	j	k
∂/∂x	∂/∂y	∂/∂z
Fx	Fy	Fz

In Maxwell's equations:

Faraday's law: ∇ × E = -∂B/∂t

The curl of the electric field is proportional to the rate of change of the magnetic field B, indicating that a changing magnetic field induces a circulating electric field (electromagnetic induction). We can imagine a bar magnet being inserted into the centre of a loop of wire: as the magnet moves into the loop a current is induced around the wire. Lenz's law tells us that the magnetic field created by the induced current resists the motion of the magnet ensuring conservation of energy (the minus sign).

Ampère-Maxwell law: ∇ × B = μ₀ J + μ₀ε₀ ∂E/∂t

The curl of the magnetic field B is related to the electric current density J and the rate of change of the electric field E. This shows how a current or a changing electric field generates a circulating magnetic field.

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Geometric and Spacetime Structure

Maxwell's equations highlight the deep connection between electric and magnetic fields. Faraday's law and Ampère's law describe how a changing electric field generates a magnetic field, and vice versa, giving these fields a dynamic interplay. This interaction shows that electric and magnetic fields are not independent; they are part of a unified structure in spacetime.

In modern physics (special relativity), this is described using the electromagnetic field tensor, where the fields are different components of the same spacetime entity F, transforming into one another depending on the observer's reference frame.

Wikipedia article

Maxwell's equations, with the help of div, grad, and curl, thus describe how electric and magnetic fields evolve and interact geometrically, revealing their unified nature in spacetime.