Questions: Electromagnetic Field Energy and Conservation

Question 1

In a simple DC circuit with a wire and a resistor, how does the electrical energy actually reach the resistor?

• A. Energy flows through the wire as drifting electrons carry kinetic energy to the resistor
• B. Energy flows through the electromagnetic field in the space surrounding the wire, not through the wire itself
• C. Energy is carried by the magnetic field alone, since no electric field exists outside a conductor
• D. Energy diffuses through the wire lattice via thermal vibrations

Answer: (B)

The Poynting vector S = (1/μ₀)(E × B) points radially inward toward a resistor carrying current, meaning electromagnetic energy flows through the surrounding space into the resistor — exactly accounting for the Joule heating rate. The wires enforce the boundary conditions for the fields, but the energy itself travels through the field outside the wire. Option A describes the common misconception; electron drift velocity is far too slow to account for the near-instantaneous energy delivery in circuits.

Question 2

What does the divergence ∇·S of the Poynting vector represent in a small volume?

• A. The rate of work done by the electromagnetic field on free charges inside the volume
• B. The magnetic energy density stored at that point
• C. The net rate at which electromagnetic energy is flowing out of that volume
• D. The total electromagnetic energy density at that point

Answer: (C)

The Poynting theorem in differential form is ∂u/∂t + ∇·S = −J·E. The divergence ∇·S is the net outward energy flux per unit volume — if ∇·S > 0, more energy is flowing out of the region than in, so the local field energy decreases. Option A describes −J·E (the work term), not the divergence. Option D describes u = ε₀E²/2 + B²/(2μ₀), the energy density itself.

Question 3

The total electromagnetic energy density at a point in free space is simply the electric energy density ε₀E²/2, since magnetic fields do not store energy.

• T. True
• F. False

Answer: False

Both electric and magnetic fields store energy. The total electromagnetic energy density is u = ε₀E²/2 + B²/(2μ₀) — the sum of the electric and magnetic contributions. This is directly analogous to how both capacitors and inductors store energy in circuits. Ignoring the magnetic term would give incorrect results for any situation involving time-varying fields or propagating waves.

Question 4

The Poynting vector for a current-carrying resistor points radially inward from the surrounding space, and its surface integral over the resistor's surface exactly equals the Joule heating rate I²R.

• T. True
• F. False

Answer: True

This is the striking consequence of Poynting's theorem applied to a resistor. A current-carrying wire has an axial E field inside (driving the current) and a circumferential B field outside (from the current). Their cross product S = (1/μ₀)E × B points radially inward toward the wire. Integrating S over the surface gives exactly the Joule heating rate — confirming that energy enters the resistor from the surrounding field, not from charge kinetic energy.

Question 5

Why does Poynting's theorem have the same mathematical structure as the charge continuity equation ∂ρ/∂t + ∇·J = 0, and what physical principle does this structural similarity express?

Model answer: Both are local conservation laws expressed as continuity equations. In Poynting's theorem (∂u/∂t + ∇·S = −J·E), u is the energy density, S is the energy flux, and −J·E is the source/sink term (work done by fields on charges). In charge continuity, ρ is the charge density, J is the charge flux, and the right side is zero (charge is strictly conserved). The structural parallel means both express the same idea: a conserved quantity can only change locally if it flows in or out (or is created/destroyed by a local source). For charge, there are no sources; for electromagnetic energy, the source term J·E accounts for the conversion between field energy and mechanical/thermal energy of charges.

This structural similarity reveals that Poynting's theorem is not just a useful formula — it is a local conservation law, meaning energy cannot teleport; it must flow continuously through space. The continuity equation form applies to any conserved quantity (charge, energy, momentum, probability in quantum mechanics). Recognizing this structure lets you interpret ∂u/∂t as the rate of change of energy stored locally, ∇·S as the divergence of the energy current, and J·E as the coupling between electromagnetic energy and matter.