8. Electromagnetic Energy, Momentum and Forces

Introduction and context

Electromagnetic fields are not merely mathematical quantities used to calculate voltage, current, or wave propagation. They store energy, transport energy through space, carry linear and angular momentum, and exert forces on matter. These properties provide the physical basis for power transmission, antennas, waveguides, optical devices, electric machines, actuators, sensors, microwave heating, and electromagnetic compatibility.

The central question of this chapter is: how can the energy and mechanical effects associated with electromagnetic fields be quantified and interpreted? Maxwell’s equations provide the field description, while Poynting’s theorem and the Maxwell stress tensor provide complementary energy and force descriptions. Poynting’s theorem concerns energy conservation and power flow. The stress tensor concerns momentum transport and the resulting force on charges, currents, and material bodies.

A field-based treatment is especially important in distributed systems. In a low-frequency circuit, power is often represented as the product of voltage and current. In a transmission line, however, the same power is physically transported by the electromagnetic fields in the space around the conductors. In an optical or microwave system, the field description is indispensable because energy and momentum may propagate through regions containing no conducting path.

Learning objectives

After studying this chapter, the reader should be able to:

1. Derive and interpret Poynting’s theorem in differential and integral forms.

2. Distinguish electromagnetic energy density, power density, stored energy, and dissipated power.

3. Explain the physical meaning of the Poynting vector in static, guided-wave, and radiating systems.

4. Formulate electromagnetic momentum density and relate it to radiation pressure.

5. Use the Maxwell stress tensor to calculate forces and torques on electromagnetic systems.

6. Apply energy and force methods to capacitors, inductors, transmission lines, actuators, and wave devices.

7. Recognize assumptions, limitations, numerical issues, and common errors in field-force calculations.

Energy stored in electromagnetic fields

For a linear, isotropic, nondispersive medium, the electric and magnetic energy densities are

\[

w_e=\frac{1}{2}\mathbf{E}\cdot\mathbf{D}=\frac{1}{2}\epsilon E^2,

\qquad

w_m=\frac{1}{2}\mathbf{B}\cdot\mathbf{H}=\frac{1}{2}\mu H^2.

\]

The total instantaneous electromagnetic energy density is therefore

\[

w=\frac{1}{2}(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H}).

\]

The total stored energy in a volume \(V\) is

\[

W=\int_V w\,dv.

\]

The electric term is familiar from a capacitor. For a capacitor of capacitance \(C\) charged to voltage \(V\),

\[

W_e=\frac{1}{2}CV^2.

\]

The magnetic term corresponds to the energy in an inductor:

\[

W_m=\frac{1}{2}LI^2.

\]

These circuit expressions are specialized forms of the field integrals. They are valid when the field distribution can be represented adequately by a lumped parameter. At higher frequencies or when dimensions become comparable with the wavelength, the distributed field description is required.

In a conducting medium, the term \(\mathbf{J}\cdot\mathbf{E}\) represents the rate at which electromagnetic energy is converted into thermal energy per unit volume. For an ohmic conductor, \(\mathbf{J}=\sigma\mathbf{E}\), so the power density is

\[

p_{loss}=\mathbf{J}\cdot\mathbf{E}=\sigma E^2=\frac{J^2}{\sigma}.

\]

For time-harmonic fields, average quantities are commonly used. With phasors defined using the convention \(e^{j\omega t}\), the time-average power density is

\[

\langle \mathbf{S}\rangle=\frac{1}{2}\operatorname{Re}(\mathbf{E}\times\mathbf{H}^*).

\]

The factor one-half results from averaging sinusoidal products over a cycle.

Poynting’s theorem: local energy conservation

Start with the curl equations in a general medium:

\[

\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},

\qquad

\nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}.

\]

Taking the scalar product of the second equation with \(\mathbf{E}\), and of the first with \(\mathbf{H}\), then subtracting, gives

\[

\mathbf{E}\cdot(\nabla\times\mathbf{H})-\mathbf{H}\cdot(\nabla\times\mathbf{E})

=\mathbf{E}\cdot\mathbf{J}+\mathbf{E}\cdot\frac{\partial\mathbf{D}}{\partial t}+\mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t}.

\]

Using the vector identity

\[

\nabla\cdot(\mathbf{E}\times\mathbf{H})

=\mathbf{H}\cdot(\nabla\times\mathbf{E})-\mathbf{E}\cdot(\nabla\times\mathbf{H}),

\]

and assuming a linear nondispersive medium, one obtains

\[

\boxed{\frac{\partial w}{\partial t}+\nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E}=0}

\]

where

\[

\mathbf{S}=\mathbf{E}\times\mathbf{H}

\]

is the Poynting vector, with units of watts per square metre.

This equation states that the rate of increase of electromagnetic energy in a small volume plus the net outward electromagnetic power flow plus the power delivered to matter is zero. It is a local statement of conservation of energy.

Integrating over a fixed volume and applying the divergence theorem gives

\[

\frac{d}{dt}\int_V w\,dv

=-\oint_A \mathbf{S}\cdot d\mathbf{a}-\int_V \mathbf{J}\cdot\mathbf{E}\,dv.

\]

The first surface integral is the net power leaving the volume. The second volume integral is the power transferred from the field to charges. If \(\mathbf{J}\cdot\mathbf{E}>0\), matter absorbs field energy. If it is negative, matter supplies energy to the field, as occurs in an active medium or generator model.

Interpreting the Poynting vector

The direction of \(\mathbf{S}\) indicates the direction of electromagnetic energy transport, while its magnitude represents power crossing a surface normal to that direction. The total power through a surface is

\[

P=\int_A \mathbf{S}\cdot d\mathbf{a}.

\]

In a uniform plane wave travelling in a lossless medium, \(\mathbf{E}\), \(\mathbf{H}\), and the propagation direction are mutually perpendicular. The magnitude relation is

\[

|\mathbf{S}|=EH=\frac{E^2}{\eta}=\eta H^2,

\]

where \(\eta=\sqrt{\mu/\epsilon}\) is the intrinsic impedance of the medium.

The Poynting vector need not be parallel to a conductor or to a conventional current. For a coaxial cable, energy primarily flows through the dielectric between the inner and outer conductors. For a rectangular waveguide, the longitudinal component of \(\mathbf{S}\) transports power even though the fields form a complex standing-wave pattern in the transverse plane. In a resonant cavity, the instantaneous Poynting vector may circulate locally while the net time-average power through the boundary is zero.

Near a radiating antenna, reactive and radiative field components coexist. Reactive fields store and return energy during each cycle, whereas the radiative component produces a nonzero average outward power flow. Consequently, a local nonzero instantaneous Poynting vector does not by itself prove that net energy is permanently radiated from a region.

Momentum carried by fields and radiation pressure

Electromagnetic energy transport is associated with momentum transport. In a simple nondispersive medium, a commonly used electromagnetic momentum density is

\[

\mathbf{g}=\frac{\mathbf{S}}{c^2}

\]

in free space, where \(c\) is the speed of light. For a plane wave, energy and momentum are related by

\[

p=\frac{U}{c},

\]

where \(U\) is the transported energy.

When electromagnetic radiation is absorbed or reflected by a surface, its momentum changes. The corresponding momentum change produces mechanical force. For normally incident radiation in free space, the radiation pressure on a perfectly absorbing surface is

\[

p_{rad}=\frac{I}{c},

\]

where \(I\) is the incident intensity. For a perfectly reflecting surface,

\[

p_{rad}=\frac{2I}{c}.

\]

For oblique incidence, only the appropriate normal component of momentum contributes to normal pressure, and tangential components may produce shear forces. Radiation pressure is small in ordinary engineering environments but becomes important in high-power lasers, solar sails, optical manipulation, vacuum systems, and precision measurement.

The treatment of electromagnetic momentum in material media requires care. The Abraham and Minkowski forms assign different canonical and kinetic momenta in matter. The apparent disagreement is resolved only by accounting consistently for material momentum and the chosen system boundary. In practical force calculations, the total momentum balance of field plus matter is the physically relevant quantity.

Maxwell stress tensor

Poynting’s theorem describes energy transfer; a corresponding momentum balance is expressed using the Maxwell stress tensor. In vacuum, its Cartesian components are

\[

T_{ij}=\epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}E^2\right)

+\frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}B^2\right),

\]

or, in dyadic notation,

\[

\mathbf{T}=\epsilon_0\left(\mathbf{E}\mathbf{E}-\frac{1}{2}E^2\mathbf{I}\right)

+\frac{1}{\mu_0}\left(\mathbf{B}\mathbf{B}-\frac{1}{2}B^2\mathbf{I}\right).

\]

The tensor relates electromagnetic momentum flux to force. For a closed surface enclosing a body, the electromagnetic force is obtained from a surface integral of the stress tensor, together with any required time-rate-of-change of field momentum inside the volume:

\[

\mathbf{F}=\oint_S \mathbf{T}\cdot\hat{\mathbf{n}}\,da

-\frac{d}{dt}\int_V \mathbf{g}\,dv,

\]

with sign conventions determined by the definition of the outward normal and momentum balance. For time-averaged steady-state problems, the average stored momentum term often vanishes, giving

\[

\langle\mathbf{F}\rangle=\oint_S \langle\mathbf{T}\rangle\cdot\hat{\mathbf{n}}\,da.

\]

The traction vector \(\mathbf{T}\cdot\hat{\mathbf{n}}\) represents force per unit area transmitted across a surface. Unlike a scalar pressure, it can have normal and tangential components. This makes the tensor suitable for calculating force and torque in nonuniform fields, electromagnetic actuators, waveguides, and particle manipulation systems.

The torque about an origin is

\[

\boldsymbol{\tau}=\oint_S \mathbf{r}\times(\mathbf{T}\cdot\hat{\mathbf{n}})\,da

-\frac{d}{dt}\int_V \mathbf{r}\times\mathbf{g}\,dv.

\]

Thus electromagnetic fields can transfer both linear and angular momentum.

Force on charges, currents, and material bodies

The local Lorentz force density on charged matter is

\[

\mathbf{f}=\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}.

\]

For a point charge, this reduces to \(\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})\). For a current-carrying conductor, the magnetic force is commonly written

\[

\mathbf{F}=\int_V \mathbf{J}\times\mathbf{B}\,dv.

\]

For a short wire segment carrying current \(I\),

\[

d\mathbf{F}=I\,d\mathbf{l}\times\mathbf{B}.

\]

The Lorentz-force method is direct when charge and current distributions are known. The stress-tensor method is often more convenient when the fields on a surrounding surface are available but the detailed internal force distribution is not.

At material interfaces, force calculations require attention to discontinuities in normal and tangential field components. Bound charges, magnetization currents, surface charges, and surface currents may all contribute. A surface enclosing the complete body and lying in a homogeneous region is often selected to avoid ambiguity associated with assigning force inside matter.

Energy methods for electromechanical forces

A second powerful approach derives force from the change in stored field energy. Suppose a mechanical coordinate \(x\) changes while electrical variables are controlled. The appropriate energy function depends on the independent variables.

For a capacitor with capacitance \(C(x)\), the stored energy at fixed charge is

\[

W_e=\frac{q^2}{2C(x)}.

\]

At fixed voltage, the electrical co-energy is more convenient:

\[

W'_e=\frac{1}{2}C(x)V^2.

\]

The force at constant voltage is

\[

F_x=\left.\frac{\partial W'_e}{\partial x}\right|_V.

\]

For a linear inductor with position-dependent inductance \(L(x)\),

\[

W_m=\frac{1}{2}L(x)I^2,

\]

and at constant current,

\[

F_x=\left.\frac{\partial W_m}{\partial x}\right|_I.

\]

These formulas express a tendency toward increased capacitance at fixed voltage and increased inductance at fixed current. They must not be applied without identifying the controlled electrical variable and the complete energy balance. A common error is to differentiate stored energy under the wrong constraint.

For nonlinear magnetic materials, the field energy is

\[

W_m=\int_V\left[\int_0^{\mathbf{B}}\mathbf{H}\cdot d\mathbf{B}\right]dv,

\]

whereas co-energy is defined through the complementary integral in the \(B-H\) relationship. The force can then be calculated by differentiating the appropriate co-energy with respect to mechanical position while holding current constant.

Applications in devices and systems

Capacitive actuators

In a variable-gap electrostatic actuator, reducing the gap increases capacitance. With applied voltage maintained by a source, the force attracts the electrodes and tends to reduce the gap further. The same principle is used in microelectromechanical systems, electrostatic switches, micromirrors, and capacitive sensors. Design must account for pull-in instability: beyond a critical displacement, electrostatic force can increase faster than the restoring mechanical force.

Magnetic actuators and electric machines

Relays, solenoids, loudspeakers, motors, and generators convert energy between electrical and mechanical forms. The magnetic force can be evaluated using Lorentz forces on conductors, Maxwell stress on an air-gap surface, or magnetic co-energy. In rotating machines, torque is associated with the variation of magnetic co-energy with angular displacement. Accurate computation must include saturation, leakage flux, fringing, eddy currents, and material hysteresis where relevant.

Transmission lines and waveguides

The power delivered by a transmission line is obtained from the surface integral of the Poynting vector. A waveguide may have conducting walls on which the tangential electric field is zero, yet energy flows down the guide through the interior fields. The stress tensor can also predict forces on waveguide walls, tuning posts, irises, and dielectric windows. In high-power systems, these forces can cause vibration, deformation, or breakdown-related failure.

Optical and microwave manipulation

Focused electromagnetic fields exert gradient forces on dielectric particles and radiation-pressure forces along the propagation direction. Optical tweezers exploit spatial field gradients to trap microscopic particles. At microwave frequencies, resonators and metamaterial structures can generate strong local field concentrations, producing measurable forces on small components. Numerical field solutions are usually coupled to mechanical models to predict displacement and stability.

Illustrative case study: force prediction in a solenoid actuator

Background

Consider an illustrative solenoid in which a movable ferromagnetic plunger enters a coil. The coil is driven by a current-controlled power converter, and the plunger position \(x\) changes the inductance \(L(x)\).

Situation and problem

The design team must estimate the holding force near the end of travel and determine whether a simplified magnetic-circuit model is reliable. Excessive force may damage the mechanical stop, while insufficient force may prevent reliable switching.

Analysis

A first model assumes a dominant air gap of length \(g\), cross-sectional area \(A\), and negligible leakage. The magnetic reluctance is approximately \(g/(\mu_0A)\), so the inductance increases as the gap decreases. At constant current, the magnetic co-energy is approximately

\[

W'_m\approx\frac{1}{2}L(x)I^2.

\]

The predicted force is

\[

F_x=\frac{1}{2}I^2\frac{dL}{dx},

\]

with the sign selected according to the coordinate direction. Alternatively, a Maxwell-stress calculation over a surface in the air gap gives an approximate magnetic pressure

\[

p_m\approx\frac{B^2}{2\mu_0},

\]

and force \(F\approx p_mA\) when the field is nearly uniform and normal to the surface.

A finite-element model is then used to include fringing, saturation of the iron, and the changing geometry. The stress tensor is integrated over a closed contour or surface in the air region surrounding the plunger. The result is compared with the co-energy derivative.

Approach and outcome

The simplified model gives a useful initial estimate and identifies the expected trend: force increases strongly as the air gap decreases. The finite-element result is lower than the unsaturated estimate near the end of travel because the iron approaches magnetic saturation and fringing alters the effective area. The final design increases the mechanical stop strength, limits the drive current, and uses a current profile that prevents impact.

Learning points

1. Energy and stress-tensor methods should agree when the same material model and boundary conditions are used.

2. A magnetic-circuit estimate is valuable for sizing but may be inaccurate near saturation and strong fringing.

3. The controlled variable matters when deriving force from energy.

4. Electromagnetic force must be evaluated together with mechanical, thermal, and reliability constraints.

Computational practice and engineering judgement

Finite-element electromagnetic software commonly computes energy, co-energy, force, torque, and stress-tensor results. Reliable use requires mesh refinement in air gaps, corners, high-gradient regions, and near material interfaces. Force obtained by differentiating total energy with respect to displacement can be sensitive to numerical noise; virtual-work methods, weighted stress tensors, or carefully chosen integration surfaces may provide better convergence.

For transient systems, instantaneous power and force can oscillate at twice the excitation frequency or at beat frequencies. Average values alone may hide vibration, acoustic excitation, fatigue, or peak-stress problems. In coupled electromagnetic-mechanical simulations, the time step must resolve both the electrical waveform and the mechanical response.

Challenges, limitations, and common mistakes

Several errors occur repeatedly in electromagnetic energy and force analysis:

• Treating \(\mathbf{E}\times\mathbf{H}\) as a universal net-power quantity without distinguishing instantaneous, average, reactive, and radiative power.
• Omitting the complex conjugate in the phasor expression for average Poynting vector.
• Using a vacuum stress tensor inside a material without considering the material model and chosen force boundary.
• Applying the fixed-voltage force formula to a fixed-charge problem, or the fixed-current magnetic formula to a fixed-flux problem.
• Integrating stress over an open surface without including all relevant surfaces or momentum terms.
• Ignoring fringing, saturation, hysteresis, displacement current, or dispersive behavior when these effects are significant.
• Confusing field energy density with energy dissipated in a conductor.
• Assuming that local field energy flow must follow the physical path of a wire.

In dispersive, lossy, anisotropic, or nonlinear media, the simple energy density expressions may require modification. Stored energy must then be defined using constitutive relations, frequency derivatives, or a full thermodynamic model. Force predictions can also depend on whether the boundary encloses only the field or both field and material contributions.

Professional, ethical, and safety considerations

Electromagnetic force and energy calculations affect equipment safety. High-field devices may experience structural failure, projectile motion of ferromagnetic objects, dielectric breakdown, arc formation, or thermal runaway. Engineering documentation should state field assumptions, material data, boundary conditions, excitation conventions, mesh verification, and uncertainty margins.

In high-power microwave, laser, and industrial heating systems, radiation exposure and unintended coupling must be assessed according to applicable safety standards and organizational procedures. Professional practice requires independent checking of critical force predictions, especially where failure could injure personnel or damage pressure vessels, rotating equipment, or high-voltage insulation.

Current developments and best practices

Modern electromagnetic design increasingly uses multiphysics simulation linking electromagnetic fields with structural deformation, heat transfer, acoustics, and control systems. Reduced-order models derived from field solutions allow real-time estimation of force and torque in digital controllers. Automatic differentiation and surrogate models can accelerate optimization, while uncertainty quantification evaluates sensitivity to material tolerances and manufacturing variation.

Best practice is to use complementary methods: derive an analytical scaling law, verify it with energy or co-energy, compare with a stress-tensor calculation, and validate the final model experimentally where feasible. Agreement between independently formulated methods is more informative than additional numerical precision in a single model.

Chapter Summary

Electromagnetic fields store energy, transport power, carry momentum, and exert force. The energy density in a linear medium is the sum of electric and magnetic contributions. Poynting’s theorem expresses local and global conservation of electromagnetic energy: field-energy change, outward power flow, and power delivered to matter must balance. The Poynting vector identifies electromagnetic power transport and is essential in waveguides, transmission lines, antennas, and resonators.

Field momentum is linked to energy flow and produces radiation pressure when waves are absorbed or reflected. The Maxwell stress tensor provides a surface-based method for calculating electromagnetic force and torque from field distributions. The Lorentz force gives a local charge-and-current description. Energy and co-energy methods provide efficient force calculations in capacitive and magnetic actuators, provided the correct controlled variable is used. Practical analysis must account for material behavior, fringing, saturation, dispersion, numerical convergence, safety, and coupled mechanical effects.

Key Terms

Poynting vector: Vector \(\mathbf{S}=\mathbf{E}\times\mathbf{H}\) representing instantaneous electromagnetic power-flow density.

Poynting’s theorem: Local and integral statement of electromagnetic energy conservation.

Electromagnetic energy density: Energy stored per unit volume in electric and magnetic fields.

Electric energy density: Stored electric energy per unit volume, commonly \(\frac12\mathbf{E}\cdot\mathbf{D}\).

Magnetic energy density: Stored magnetic energy per unit volume, commonly \(\frac12\mathbf{B}\cdot\mathbf{H}\).

Power density: Rate of energy transfer per unit area or volume, depending on context.

Reactive power flow: Oscillatory energy exchange that has zero net average transfer over a cycle.

Radiation pressure: Mechanical pressure produced by transfer of electromagnetic momentum.

Electromagnetic momentum density: Momentum carried per unit volume by an electromagnetic field.

Maxwell stress tensor: Tensor describing electromagnetic momentum flux and mechanical traction.

Lorentz force density: Force per unit volume \(\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}\).

Traction: Force per unit area acting across a surface.

Co-energy: Complementary energy function useful for force calculation under fixed-current or fixed-voltage conditions.

Virtual work: Force-calculation method based on the change of field energy under an infinitesimal displacement.

Intrinsic impedance: Ratio of electric-field magnitude to magnetic-field magnitude in a plane wave.

Radiative field: Field component that carries net average energy away from a source.

Reactive field: Field component that stores and returns energy locally during oscillation.

Magnetic pressure: Approximate normal stress \(B^2/(2\mu_0)\) in a uniform air-gap field.

Momentum flux: Rate at which electromagnetic momentum crosses a surface.

Review Questions

Short Answer Questions

1. State Poynting’s theorem in differential form and identify each term.

2. What is the physical meaning of the Poynting vector?

3. Define radiation pressure for an absorbing surface in free space.

4. Write the Lorentz force density equation.

5. Why is co-energy useful in electromechanical force calculations?

Long Answer Questions

1. Derive the differential and integral forms of Poynting’s theorem from Maxwell’s curl equations.

2. Explain how electromagnetic momentum leads to radiation pressure on absorbing and reflecting surfaces.

3. Describe the Maxwell stress tensor and explain how it is used to calculate force and torque.

4. Compare Lorentz-force, stress-tensor, and energy-based methods for calculating actuator force.

5. Discuss the interpretation of power flow in a coaxial cable, waveguide, and resonant cavity.

Analytical / Application Questions

1. A plane wave in free space has electric-field amplitude \(E_0\). Derive its average power density and radiation pressure on a perfectly absorbing surface.

2. A variable capacitor has capacitance \(C(x)\) and is connected to an ideal constant-voltage source. Derive the force as a function of position and explain its direction.

3. A solenoid has inductance \(L(x)\) that increases as a plunger moves inward. Determine the force under constant current and discuss how saturation would alter the result.

22. Multiple Choice Questions with Answers

Question 1

Which equation expresses local electromagnetic energy conservation in a linear, nondispersive medium?

A. ∇·E=ρ/ε

B. ∂w/∂t+∇·S+J·E=0

C. ∇×E=J

D. ∂ρ/∂t+∇×J=0

Correct Answer: B. ∂w/∂t+∇·S+J·E=0

Question 2

The direction of the instantaneous Poynting vector represents the direction of

A. Charge motion only

B. Magnetic flux

C. Electromagnetic energy transport

D. Mechanical displacement

Correct Answer: C. Electromagnetic energy transport

Question 3

For a perfectly absorbing surface normally illuminated by intensity I in free space, the radiation pressure is

A. I/c

B. 2I/c

C. Ic

D. I/c²

Correct Answer: A. I/c

Question 4

The local force density on charged matter is given by

A. ρH+J×D

B. ρE+J×B

C. εE²+μH²

D. E×H

Correct Answer: B. ρE+J×B

Question 5

For a harmonic field using phasors, the average Poynting vector is

A. Re(E×H)

B. E×H*

C. (1/2)Re(E×H*)

D. (1/2)Im(E×H)

Correct Answer: C. (1/2)Re(E×H*)

Question 6

At constant voltage, the force on a variable capacitor is obtained from

A. The derivative of capacitance co-energy with respect to position

B. The derivative of resistance with respect to current

C. The magnetic flux divided by voltage

D. The divergence of magnetic field

Correct Answer: A. The derivative of capacitance co-energy with respect to position

Question 7

Which method is especially suitable when fields are known on a closed surface surrounding a body but internal current details are unavailable?

A. Kirchhoff voltage law

B. Maxwell stress tensor

C. Ohm’s law alone

D. Faraday’s law alone

Correct Answer: B. Maxwell stress tensor

Question 8

For a plane wave in a lossless medium, E, H, and the propagation direction are

A. Parallel

B. Mutually perpendicular

C. Randomly oriented

D. Antiparallel in all cases

Correct Answer: B. Mutually perpendicular

Question 9

A nonzero instantaneous Poynting vector in a resonant cavity necessarily means that

A. There is permanent net power leaving the cavity

B. The cavity is dissipating all stored energy

C. Energy may be circulating or exchanged locally without net average output

D. The electric field is zero

Correct Answer: C. Energy may be circulating or exchanged locally without net average output

Question 10

In a magnetic actuator operated at constant current, the linear force is commonly calculated from

A. The derivative of magnetic co-energy with respect to position

B. The time derivative of resistance

C. The divergence of electric flux only

D. The square root of stored electric energy

Correct Answer: A. The derivative of magnetic co-energy with respect to position