Radiation from a Dipole Questions and Answers

Q: What is a retarded potential?

Because the field takes time to travel, the potential at a point now depends on the source as it was one propagation time earlier, and writing the source at that retarded time is what turns the static potential integrals into radiating ones. Every radiation result follows from this single modification. Ohmic loss, transient decay and reactive phase lag are all real effects but none of them is what the word retarded refers to here.

Q: What is a Hertzian dipole?

The Hertzian or elementary dipole is an idealised element so short that the current may be taken as constant along it, and it serves as the building block from which any wire antenna is assembled by integration. Real short antennas have a current tapering to zero at the ends, which is why their effective length is half the physical length. A half wave dipole, an end fed wire and a crossed pair are all specific practical antennas rather than this abstraction.

Q: How do the field terms of a Hertzian dipole vary with distance?

The exact solution contains terms in one over r, one over r squared and one over r cubed, and only the first survives to make up the far field. The inverse square term is the induction field and the inverse cube term the electrostatic field, both of which dominate close in and vanish quickly. The distance at which the radiation term overtakes the others is a fraction of a wavelength, which is why the reactive near field is so shallow in wavelengths even for a large antenna.

Q: Which field term is responsible for radiated power?

Only the inverse first power term gives a Poynting vector falling as the inverse square of range, so that the power through a sphere is independent of its radius, which is exactly what radiation means. The other terms carry energy outward and back again within a cycle and transport no net power. That is the cleanest way to distinguish radiation from stored energy, and it explains why the far field is defined by the dominance of a single term, whatever the antenna.

Q: What does the Poynting vector represent?

The Poynting vector is the cross product of the electric and magnetic fields and gives the instantaneous power crossing unit area, in watts per square metre. Integrating its time average over a closed surface gives the radiated power, which is how radiation resistance is derived. It points in the direction of propagation, perpendicular to both fields. Energy density and intrinsic impedance are related quantities but are computed differently and mean different things.

Q: The radiation resistance of a short dipole is \( 80\pi^{2}(dl/\lambda)^{2} \). What is it for dl = 0.05 lambda?

Substituting the length ratio directly:\[ R_{r} = 80\pi^{2}(0.05)^{2} \approx 1.97 \]giving about 1.97 ohms. Against a conductor and matching loss of even one ohm the efficiency would be barely two thirds, and at a tenth of this length the resistance falls a hundredfold. The square law is the quantitative reason electrically small antennas are inefficient, and it is worth being able to evaluate quickly.

Q: How does the radiation resistance of a short dipole vary with its length?

Radiation resistance rises as the square of the length measured in wavelengths, so halving a short antenna quarters its radiation resistance while its loss resistance falls barely at all. That is the whole difficulty of small antennas expressed in one relation. The same square law means that raising the frequency of a fixed physical antenna improves its radiation resistance rapidly, which is why a whip that is hopeless at one frequency can be efficient an octave higher.

Q: What current distribution is normally assumed along a thin dipole?

The standard thin wire approximation takes a sinusoidal standing wave of current that must vanish at the open ends, since no current can flow into free space there. For a half wave dipole this puts the maximum at the centre, which is why centre feeding gives a convenient impedance. A uniform distribution belongs to the idealised Hertzian element, and the other two shapes describe no physical wire antenna. The assumption is remarkably accurate for thin conductors.

The radiation resistance of a Hertizian dipole antenna of length \(\frac{\lambda}{60}\) is

A.0.493 Ohm
B.0.22 Ohm
C.2.2 Ohm
D.22 Ohm

Exam Year(s)

UGC NET - 2012,

Answer: B

The radiation resistance of a short dipole is \(80\pi^2 (l/\lambda)^2\), which for \(l = \lambda/60\) is about 0.22 ohm (option B).

\[ R_{rad} = 80\pi^2\left(\frac{1}{60}\right)^2 \]

\[ R_{rad} = \frac{789.6}{3600} \approx 0.22\ \Omega \]

The value 0.493 ohm (A) would belong to a longer dipole, near \(\lambda/40\). The values 2.2 and 22 ohm are 10 and 100 times too big, as if the length were \(\lambda/19\) or \(\lambda/6\).

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The value of radiation resistance of a Hertizian dipole of length \(\frac{\lambda}{80}\) is given by

A.0.493 Ω
B.0.219 Ω
C.12.3 Ω
D.0.123 Ω

Exam Year(s)

UGC NET - 2014,

Answer: D

The radiation resistance is 0.123 ohm. For a short (Hertzian) dipole of length \(l\) the radiation resistance is \(R_{r} = 80\pi^{2}\left(\frac{l}{\lambda}\right)^{2}\). With \(l = \lambda/80\):

\[ R_{r} = 80\pi^{2}\times\frac{1}{6400} \]

\[ R_{r} = \frac{\pi^{2}}{80} \]

\[ R_{r} \approx \frac{9.87}{80} = 0.123\ \Omega \]

The values 0.493, 0.219 and 12.3 ohm do not follow from this formula for \(l/\lambda = 1/80\); 12.3 is just 100 times the correct figure.

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What is a retarded potential?

A.A potential evaluated at an earlier time to allow for propagation delay
B.A potential reduced by the resistance of the conductor
C.A potential measured after the transient has decayed away
D.A potential that lags the current by ninety degrees in phase

Answer: A

Because the field takes time to travel, the potential at a point now depends on the source as it was one propagation time earlier, and writing the source at that retarded time is what turns the static potential integrals into radiating ones. Every radiation result follows from this single modification. Ohmic loss, transient decay and reactive phase lag are all real effects but none of them is what the word retarded refers to here.

Discuss in Forum

What is a Hertzian dipole?

A.Two dipoles crossed at right angles and fed in quadrature
B.A current element much shorter than a wavelength with uniform current
C.A dipole exactly one half wavelength long
D.A dipole fed at one end rather than at its centre

Answer: B

The Hertzian or elementary dipole is an idealised element so short that the current may be taken as constant along it, and it serves as the building block from which any wire antenna is assembled by integration. Real short antennas have a current tapering to zero at the ends, which is why their effective length is half the physical length. A half wave dipole, an end fed wire and a crossed pair are all specific practical antennas rather than this abstraction.

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How do the field terms of a Hertzian dipole vary with distance?

A.As the inverse first and second powers only
B.As the inverse square and inverse fourth powers
C.As the inverse first, second and third powers of range
D.All terms vary as the inverse square of range

Answer: C

The exact solution contains terms in one over r, one over r squared and one over r cubed, and only the first survives to make up the far field. The inverse square term is the induction field and the inverse cube term the electrostatic field, both of which dominate close in and vanish quickly. The distance at which the radiation term overtakes the others is a fraction of a wavelength, which is why the reactive near field is so shallow in wavelengths even for a large antenna.

Discuss in Forum

Which field term is responsible for radiated power?

A.The term varying as one over the range squared
B.The term varying as one over the range cubed
C.All three terms contribute equally to the far field
D.The term varying as one over the range

Answer: D

Only the inverse first power term gives a Poynting vector falling as the inverse square of range, so that the power through a sphere is independent of its radius, which is exactly what radiation means. The other terms carry energy outward and back again within a cycle and transport no net power. That is the cleanest way to distinguish radiation from stored energy, and it explains why the far field is defined by the dominance of a single term, whatever the antenna.

Discuss in Forum

What does the Poynting vector represent?

A.Power flow per unit area in an electromagnetic field
B.The direction of the electric field vector by itself
C.The stored energy density in the field
D.The impedance presented by the medium to a wave

Answer: A

The Poynting vector is the cross product of the electric and magnetic fields and gives the instantaneous power crossing unit area, in watts per square metre. Integrating its time average over a closed surface gives the radiated power, which is how radiation resistance is derived. It points in the direction of propagation, perpendicular to both fields. Energy density and intrinsic impedance are related quantities but are computed differently and mean different things.

Discuss in Forum

The radiation resistance of a short dipole is \( 80\pi^{2}(dl/\lambda)^{2} \). What is it for dl = 0.05 lambda?

A.About 39.5 ohms, twice the dipole value
B.About 1.97 ohms
C.About 19.7 ohms
D.About 0.20 ohms

Answer: B

Substituting the length ratio directly:

\[ R_{r} = 80\pi^{2}(0.05)^{2} \approx 1.97 \]

giving about 1.97 ohms. Against a conductor and matching loss of even one ohm the efficiency would be barely two thirds, and at a tenth of this length the resistance falls a hundredfold. The square law is the quantitative reason electrically small antennas are inefficient, and it is worth being able to evaluate quickly.

Discuss in Forum

How does the radiation resistance of a short dipole vary with its length?

A.As the cube of the length in wavelengths
B.Inversely with the length in wavelengths
C.As the square of the length in wavelengths
D.In direct proportion to the length in wavelengths

Answer: C

Radiation resistance rises as the square of the length measured in wavelengths, so halving a short antenna quarters its radiation resistance while its loss resistance falls barely at all. That is the whole difficulty of small antennas expressed in one relation. The same square law means that raising the frequency of a fixed physical antenna improves its radiation resistance rapidly, which is why a whip that is hopeless at one frequency can be efficient an octave higher.

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10.

What current distribution is normally assumed along a thin dipole?

A.Uniform along the whole length of the wire
B.Triangular with a peak at each end
C.Exponentially decaying from the feed point
D.Sinusoidal, going to zero at the open ends

Answer: D

The standard thin wire approximation takes a sinusoidal standing wave of current that must vanish at the open ends, since no current can flow into free space there. For a half wave dipole this puts the maximum at the centre, which is why centre feeding gives a convenient impedance. A uniform distribution belongs to the idealised Hertzian element, and the other two shapes describe no physical wire antenna. The assumption is remarkably accurate for thin conductors.

Discuss in Forum

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Radiation from a Dipole Questions and Answers

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