Illustrated Physics
physics, drawn
Concept

Maxwell stress — where it appears

The tensor giving the force per unit area an electromagnetic field transmits across a surface. Its integral over a closed surface is the force on everything inside, whatever the contents are, and it reads as a tension along the field lines and a pressure across them, both equal to the field's energy density.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The only part of the field that can pull. A dielectric slab part-way into a parallel-plate capacitor. Everywhere except at the slab's edge the field is perpendicular to the plates and therefore perpendicular to the direction the slab can move, so it exerts no force along that direction at all: inside the parallel-plate model, which is uniform between the plates and zero outside them, nothing pulls the slab anywhere. The force lives in the bowed lines drawn at the edge, where the field leaks past the end of the dielectric and acquires a component along the plates. Those lines are what the model throws away as a small correction near the boundary, and they are the entire mechanism. The energy method sidesteps the drawing altogether: differentiate the total energy with respect to the insertion and the answer is 1.328e-3 newtons, inward, without ever asking where on the slab the force is applied.

The force that lives where the model is not

A slab of glass held at the mouth of a charged capacitor is pulled in. Inside the parallel-plate model there is no force at all — the field is perpendicular to the slab's motion everywhere — and the same model's energy nevertheless gives the pull exactly right. The mechanism is entirely in the part of the field the model throws away.

electromagnetism · Dielectrics What the plane between them carries. The stress transmitted across the plane halfway between two charges of 10 nC held 1 cm apart, against distance from the axis in units of the half-separation. For two like charges the field on that plane lies entirely in it — checked here rather than assumed — so the plane sees only the pressure across the lines, the stress is negative everywhere, and the two halves are pushed apart. For opposite charges the field on the plane is entirely perpendicular to it, the plane sees only the tension along the lines, the stress is positive, and the halves are pulled together. Faraday's two words for a field line, tension along it and pressure across it, are exactly these two curves; the whole content of the tensor is that they are the same quantity, ε₀E²/2, wearing two signs. The force is what is left after integrating either curve over the plane, and both integrals come to the same magnitude — the Coulomb force — which is the next figure.

The force read off a surface that touches nothing

Draw any closed surface through empty space, measure the field on it, and the sum of one expression over that surface is the total force on everything inside — whatever the contents are, and without knowing anything about them. The expression is Maxwell's stress tensor, and it turns Faraday's guess about tension along a field line into an exact statement.

electromagnetism · Field energy One magnitude, and a direction that turns the wrong way. The force per unit area a magnetic field of 0.01 tesla exerts on a surface, drawn for surface normals at 0°, 30°, 45°, 60°, 90° to the field. The short grey arrows are the normals; the long coloured ones are the tractions. Every traction has the same length, 40 pascals, because the magnitude of T·n does not depend on the orientation of n at all — and the direction is the normal reflected in the field rather than carried round with it, so as the surface turns one way the force turns the other. A surface cutting across the field is pushed (the magnetic pressure); a surface cutting along it is pulled (the magnetic tension); and at 45° the traction lies in the surface, which is a shear and is neither.

The same force whichever way the surface faces

A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.

astrophysics · Flux freezing

Named alongside it

The objects these essays reach for when they reach for this one.

BatteryBound chargeBuoyancyCapacitanceConservationDielectricDielectrophoresisElectric fieldElectrostatic forceEnergy densityEnergy methodEquilibrium

All concepts