The Poynting vector is defined as
Answer: A
The cross product of the two field intensities has the units of watts per square metre and represents the power crossing unit area. Its surface integral over a closed surface is the total power leaving the enclosed volume.
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The units of the Poynting vector are
Answer: B
Correct answer: watts per square metre
\( \mathbf{S} = \mathbf{E}\times\mathbf{H} \) has units of \( \mathrm{V/m} \times \mathrm{A/m} = \mathrm{VA/m^2} = \mathrm{W/m^2} \). It is a power *density*, not a power, which is the distinction the fourth option trades on - the total power is obtained only by integrating it over a surface. Joules per cubic metre is the energy density, a separate quantity in the same theorem.
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Poynting's theorem is a statement of
Answer: C
It says that the power entering a volume through its surface equals the rate of increase of the energy stored within plus the power dissipated there. Every term follows from Maxwell's equations with no additional assumption.
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The theorem is obtained from Maxwell's equations by forming
Answer: D
Expanding \( \nabla \cdot (\mathbf{E} \times \mathbf{H}) \) by the vector identity and substituting the two curl equations produces the three terms of the theorem directly. It is one of the tidiest derivations in the subject.
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The three terms of Poynting's theorem for a volume are the surface power flow, the rate of change of stored energy and
Answer: A
The dissipation term is the volume integral of \( \sigma E^2 \), which is Joule heating expressed in field quantities. In a lossless region it vanishes and the theorem reduces to a balance between inflow and stored energy.
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For a sinusoidally varying field the time-averaged Poynting vector is
Answer: B
Using the conjugate of one field picks out the in-phase component, and the factor of one half converts from peak to mean when the amplitudes are peak values. The imaginary part represents reactive power that flows to and fro without net transfer.
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The imaginary part of the complex Poynting vector represents
Answer: C
It corresponds to energy surging into a region and back again each half cycle, exactly as reactive power does in a circuit. Only the real part carries net energy, and near an antenna the reactive part dominates while far away only the real part survives.
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The instantaneous Poynting vector of a plane wave in a lossless medium
Answer: D
With the two fields in phase their product is proportional to the square of a sinusoid, which pulsates between zero and twice its mean at double the frequency but never goes negative. Its mean is the power the wave carries.
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In a coaxial cable carrying direct current to a load, the energy travels
Answer: A
The Poynting vector is axial and non-zero only where both a radial electric field and a circumferential magnetic field exist, which is the dielectric. Integrating it over the annular cross-section gives exactly the product of the voltage and the current.
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Inside the copper of a current-carrying conductor the Poynting vector points
Answer: B
The small axial electric field within the copper crossed with the circumferential magnetic field gives a vector directed into the conductor, and its surface integral is exactly the \( I^2R \) loss. The energy that heats a wire arrives through its surface from the field.
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