EC-2.1 Vector Calculus and the Static Electric Field

What this is and why it exists

Most of the difficulty students report with electromagnetics is difficulty with the notation, not with the physics.

So this topic builds the notation first. Then it spends that notation on the static electric field. Those results are already familiar from earlier physics, so they can be checked against what you expect.

The vocabulary

• Cartesian, cylindrical and spherical coordinates — three ways of naming the same points in space.
• Gradient — an operation on a scalar that gives the direction in which it increases fastest.
• Divergence — an operation on a field that measures how much is flowing out of a point.
• Curl — an operation on a field that measures its circulation around a point.
• Divergence theorem — converts a volume integral of divergence into a surface integral.
• Stokes' theorem — converts a surface integral of curl into a line integral around the edge.
• Coulomb's law — the force between two charges, falling with the square of the distance.
• Gauss's law — the flux out of a closed surface equals the charge it encloses.
• Potential — a single scalar from which the static electric field can be recovered.
• Dielectric — an insulating material that polarises and so reduces the field inside it.

The mental model

Coordinates come first because the choice is free and it changes everything. A long straight wire has cylindrical symmetry. A point charge has spherical symmetry. Pick the system that matches and the problem shrinks to one variable. Pick the wrong one and the same problem becomes three. Converting a vector between systems is mechanical work, and it is worth practising until it is automatic.

Then the three operations. Attach a meaning to each and the equations later read as sentences rather than as symbols.

The gradient acts on a scalar and produces a vector. It points the way the scalar climbs fastest, and its size says how steeply. The divergence acts on a vector and produces a scalar. It asks whether more is flowing out of a tiny region than into it. That is a way of asking whether a source sits at that point. The curl acts on a vector and produces a vector. It asks whether the field circulates around a point, and it points along the axis of that circulation.

The two integral theorems are the ones that earn their time later. The divergence theorem says that adding up the divergence throughout a volume gives the same answer as measuring the flow out through its surface. Stokes' theorem says that adding up the curl over a surface gives the same answer as walking round its edge. Those two conversions are exactly what turns Maxwell's equations from their point form into their region form, and back.

Now spend the notation. Coulomb's law adds up the contribution of every charge. It always works and it is laborious. Gauss's law encloses the charge in a surface and relates the flux crossing it to the charge inside. It takes one line, and only when the symmetry makes the field uniform over that surface. Recognising which situation you are in is the whole judgement here.

Potential is the labour-saving move. The static electric field is conservative, which means the work done moving a charge between two points does not depend on the path. So one scalar carries all the information, and scalars are far easier to compute with than vectors. Recover the field as the gradient of the potential, with a minus sign. Say the minus sign aloud rather than memorising it: the field points from high potential to low, and the gradient points the other way.

Dielectrics close the topic. Put an insulator in a field and its molecules line up slightly, producing a field of their own that opposes the applied one. The field inside is therefore weaker, so more charge is needed for the same voltage. That is what capacitance measures.

At an interface between two materials, two rules carry across. The tangential electric field is the same on both sides. The normal component of flux density is the same too, when no free charge sits on the surface. Those two rules solve most interface problems on their own.

What you should now be able to explain or do

Choose the coordinate system a symmetry hands you, and convert a vector between systems. Say in words what gradient, divergence and curl each measure. State both integral theorems and what each lets you exchange. Choose between adding charges and enclosing them, and justify the choice. Find capacitance for a standard arrangement, and apply the two boundary rules at an interface.

Check yourself