Questions: Energy Stored in Electric and Magnetic Fields

Question 1

According to the electric energy density formula u_E = ½ε₀E², what happens to the energy stored in a region of electric field if the field strength is doubled?

• A. The stored energy doubles, because energy is proportional to field strength
• B. The stored energy increases by a factor of 4, because energy is proportional to E²
• C. The stored energy increases by a factor of √2, because energy density is the square root of field strength
• D. The stored energy is unchanged if the volume of the field region does not change

Answer: (B)

Energy density is u_E = ½ε₀E², so it scales as the square of the field. Doubling E replaces E² with (2E)² = 4E², quadrupling the energy density — and therefore the total stored energy in any fixed volume. This nonlinearity has practical consequences: a capacitor charged to twice the voltage stores four times the energy, not twice. The quadratic dependence on field strength (or voltage) is a fundamental feature of field energy, not a curiosity.

Question 2

Where is the energy stored in a charged parallel-plate capacitor?

• A. In the electric charges on the surface of the plates
• B. In the electric potential difference measured across the terminals
• C. In the electric field distributed throughout the volume between the plates
• D. Equally split between the conductor material of the plates and the space between them

Answer: (C)

The deep insight is that the field itself is a physical entity that stores energy. The energy is not 'in the charges' or 'at the terminals' — it is spread throughout the space where the field exists. This is made concrete by the derivation: U = ½CV² = ½ε₀E² × (volume between the plates). The quantity (volume) × (energy density) gives the total energy, and the energy density ½ε₀E² is uniform between ideal parallel plates. This distinction matters for electromagnetic waves: a wave propagates through space carrying energy in its oscillating fields, with no charges present at all.

Question 3

Doubling the electric field strength in a region doubles the electric energy stored in that region.

• T. True
• F. False

Answer: False

The electric energy density is u_E = ½ε₀E², which is proportional to E squared, not E. Doubling E quadruples u_E (and therefore the total stored energy in that volume). This is the same nonlinearity that makes capacitors store four times the energy at twice the voltage. A student who memorizes 'energy is related to the field' but misses the squared dependence will consistently underestimate how rapidly field energy grows.

Question 4

In an electromagnetic wave propagating through vacuum, the electric and magnetic fields carry equal energy densities at every point.

• T. True
• F. False

Answer: True

In a plane wave in vacuum, u_E = ½ε₀E² and u_B = B²/(2μ₀) are equal at every point and moment. This is not a coincidence — it follows from the relationship between E and B in a wave (E = cB, with c = 1/√(ε₀μ₀)) and the symmetry of Maxwell's equations. The total energy flux (Poynting vector) is carried equally by both field components. This result makes sense conceptually: neither field can sustain itself without the other, so they contribute equally to the wave's energy.

Question 5

Why is it physically meaningful to say that energy is 'stored in the electromagnetic field' rather than 'stored in the capacitor plates' or 'stored in the circuit'? What conceptual work does this distinction do?

Model answer: Saying energy resides in the field rather than the plates or wires means that the field is a physical entity in its own right — one that can carry energy through space even in the absence of charges or conductors. This becomes essential for electromagnetic waves: a radio wave propagates through empty space and delivers energy to a distant antenna, with no physical connection between transmitter and receiver. If energy were stored only in charges or conductors, this would be impossible to explain. The field-energy formulation allows you to calculate how much energy a wave carries using u_E and u_B integrated over the wave's volume.

This distinction is also the conceptual bridge to the Poynting vector and the full theory of electromagnetic radiation. It represents a shift from 'action at a distance' thinking (charges reach out and affect other charges) to field thinking (charges create fields, fields carry energy and momentum, fields interact with other charges). The field becomes the primary physical reality, not just a bookkeeping device.