Questions: Poynting Theorem and Energy Conservation

Question 1

A battery is connected to a resistor by copper wires. According to Poynting's theorem, how does energy actually travel from the battery to the resistor?

• A. Along the wire, carried by the directed drift of conduction electrons
• B. Through the electromagnetic fields surrounding the wire, flowing radially inward toward the resistor
• C. Entirely within the resistor, where electrical potential energy converts to heat
• D. Along the surface of the wire in the direction of conventional current

Answer: (B)

This is the most counterintuitive result of Poynting's theorem. Outside a current-carrying wire connected to a battery, the electric field E points along the wire (from high to low potential) while the magnetic field B curls around the wire. Their cross product E×B — the Poynting vector S — points radially inward, toward the wire axis. Energy flows from the surrounding space into the resistor, not along the wire itself. The wire guides the electromagnetic fields; the fields carry the energy through the space around the wire.

Question 2

In the Poynting theorem ∂u/∂t + ∇·S = −J·E, what physical quantity does the term J·E represent?

• A. The rate of electromagnetic energy flowing across the boundary of a volume
• B. The rate of change of stored electromagnetic field energy density
• C. The power delivered per unit volume to charges — the rate at which field energy converts to mechanical or thermal energy
• D. The net electromagnetic energy stored in a bounded region

Answer: (C)

J·E is current density dotted with electric field, which has units of W/m³ — power per unit volume. It represents the rate at which the electromagnetic field does work on charges, converting field energy into kinetic or thermal energy in matter (e.g., Ohmic heating). The negative sign on the right side means that when J·E is positive (field doing work on matter), the field energy in the region must be decreasing or flowing outward. The other terms account for storage (∂u/∂t) and transport (∇·S).

Question 3

The Poynting vector S = (1/μ₀)(E×B) can be derived directly from Maxwell's equations using only algebra, without postulating any additional assumptions about energy.

• T. True
• F. False

Answer: True

This is a crucial feature of Poynting's theorem — it is not an independent postulate but a mathematical consequence of Maxwell's equations. The derivation proceeds by taking E·(Ampère-Maxwell) − B·(Faraday), applying the vector identity ∇·(E×B) = B·(∇×E) − E·(∇×B), and rearranging. No new physics is introduced; the energy conservation law is already encoded within Maxwell's equations, and the theorem makes it explicit.

Question 4

In a resistor carrying a steady current, the Poynting vector inside the resistor points in the direction of current flow.

• T. True
• F. False

Answer: False

Inside a resistor with steady current, E points in the direction of current (along the wire axis) and B curls around the current. Their cross product E×B — the Poynting vector — therefore points radially inward, perpendicular to the current direction. This inward-pointing energy flux is exactly what delivers power to the resistor: electromagnetic energy flows in from the surrounding space and is converted to heat. A Poynting vector aligned with current flow would represent energy moving along the wire, which is not what happens.

Question 5

Why does the Poynting vector show that electromagnetic energy flows through the space surrounding a wire rather than through the wire itself? Describe the geometry of E and B outside a current-carrying wire and what their cross product implies.

Model answer: Outside a straight wire carrying current in a circuit with a battery, two fields coexist: an electric field E pointing along the wire (from higher to lower potential, in the direction of conventional current) and a magnetic field B that circles around the wire (by Ampère's law). The Poynting vector S = (1/μ₀)(E×B) is perpendicular to both — which means it points radially inward toward the wire axis. This inward energy flux is what delivers electromagnetic energy from the fields to the wire and ultimately to the resistor. The wire is not a pipe for energy; it is a guide for the fields that carry energy through the surrounding space.

This result challenges the intuitive 'pipe' picture of circuits where current carries energy along the wire. In reality the conduction electrons have very little kinetic energy; almost all the energy being transferred is in the electromagnetic field configuration outside the conductor. Poynting's theorem makes this precise and quantitative: integrate the inward Poynting flux over the surface of any segment of wire or resistor and you get exactly the power being delivered to that segment.