Uniform Plane Waves Questions and Answers

Q: Consider the following statements about a plane wave in a lossless medium.1. The two fields are in phase 2. The wave is transverse in both fields 3. The intrinsic impedance is complex

Correct answer: 1 and 2 only are correctStatements 1 and 2 are correct: in a lossless medium the electric and magnetic fields are exactly in phase, and both lie transverse to the direction of travel. Statement 3 is wrong - \( \eta = \sqrt{\mu/\epsilon} \) is purely real when there is no loss, and it is exactly that reality which puts the two fields in phase. A complex impedance is the signature of a lossy medium.

Q: Consider the following statements about reflection at normal incidence.1. Equal impedances give no reflection 2. A perfect conductor gives a coefficient of minus one 3. The transmitted amplitude is one minus the reflection coefficient

Correct answer: 1 and 2 only are correctStatements 1 and 2 are correct: matched impedances give no reflection, and a perfect conductor gives \( \Gamma = -1 \). Statement 3 has the sign wrong - the transmission coefficient is \[ \tau = 1 + \Gamma, \] not \( 1-\Gamma \). The relation looks odd until one notices that the tangential electric field must be continuous, so the transmitted amplitude equals the incident plus the reflected.

41.

True or false: the reflection coefficient at a boundary can exceed unity in magnitude.

A.False for a passive boundary, the reflected power then exceeding the incident
B.True, whenever the second medium is a conductor
C.True, whenever the second impedance exceeds the first
D.True, the coefficient having no upper limit

Answer: A

Correct answer: False for a passive boundary, the reflected power then exceeding the incident

A passive boundary cannot return more power than arrives, and since the reflected power goes as \( |\Gamma|^2 \), a magnitude above one would violate conservation of energy. The coefficient can be negative, as at a perfect conductor where it is exactly \( -1 \), and it can be complex for a lossy second medium - but its magnitude never exceeds unity.

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42.

Assertion: a wave is totally reflected from a perfect conductor. Reason: the tangential electric field must vanish at its surface.

A.the assertion is false although the reason offered is a true statement in itself
B.both the assertion and the reason are true and the reason explains the assertion
C.both the assertion and the reason are true but the reason does not explain the assertion
D.the assertion is true but the reason given for it is false

Answer: B

The only way to satisfy the boundary condition is to add a reflected wave whose tangential field cancels the incident one at the surface, and that requires a coefficient of minus one. The condition and the total reflection are two aspects of the same fact.

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43.

Assertion: the group velocity in a waveguide is less than the speed of light. Reason: the phase velocity in a waveguide exceeds the speed of light.

A.the assertion is true but the reason given for it is false
B.the assertion is false although the reason offered is a true statement in itself
C.both the assertion and the reason are true and the reason explains the assertion
D.both the assertion and the reason are true but the reason does not explain the assertion

Answer: C

In a simple waveguide the product of the two velocities is the square of the free space speed, so one exceeding it forces the other below. The reason therefore does account for the assertion, and relativity constrains only the group velocity.

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44.

Assertion: a quarter-wave layer of the geometric mean impedance eliminates reflection. Reason: it transforms the load impedance into the source impedance.

A.both the assertion and the reason are true but the reason does not explain the assertion
B.the assertion is true but the reason given for it is false
C.the assertion is false although the reason offered is a true statement in itself
D.both the assertion and the reason are true and the reason explains the assertion

Answer: D

A quarter-wave section presents the square of its own impedance over the load, so with the geometric mean the transformed value equals the source and the boundary is matched. It works only at the frequency for which the layer is a quarter wavelength thick.

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45.

Match the medium with the behaviour of a wave in it.

1. Lossless dielectric 2. Good conductor 3. Free space

P. Impedance of 377 ohms Q. Attenuation equal to the phase constant R. No attenuation, impedance real and below 377 ohms

A.1-R, 2-Q, 3-P
B.1-Q, 2-R, 3-P
C.1-R, 2-P, 3-Q
D.1-P, 2-Q, 3-R

Answer: A

The three cases correspond to the three values the ratio of conduction to displacement current can take, and each has its own characteristic impedance and attenuation. Recognising which applies is the first step in any propagation calculation.

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46.

Match the phase difference between two perpendicular equal components with the polarisation it gives.

1. Zero 2. Ninety degrees 3. Forty-five degrees

P. Elliptical Q. Linear R. Circular

A.1-P, 2-R, 3-Q
B.1-Q, 2-R, 3-P
C.1-R, 2-Q, 3-P
D.1-Q, 2-P, 3-R

Answer: B

The three are points on one continuum: as the phase difference grows from zero to ninety degrees the line opens into an ellipse and then closes into a circle. Unequal amplitudes give an ellipse at every phase difference except zero.

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47.

The steps of finding the reflected and transmitted fields at a boundary are listed below. Put them in order.

1. Compute the reflection coefficient 2. Find the intrinsic impedance of each medium 3. Compute the standing wave ratio 4. Obtain the transmitted amplitude from one plus the coefficient

A.2, 4, 1, 3
B.2, 1, 3, 4
C.2, 1, 4, 3
D.1, 2, 4, 3

Answer: C

The impedances come first because everything else is computed from them, then the reflection coefficient, then the transmission coefficient which is one plus it, and the standing wave ratio follows from the magnitude of the reflection coefficient last.

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48.

Consider the following statements about a plane wave in a lossless medium.

1. The two fields are in phase 2. The wave is transverse in both fields 3. The intrinsic impedance is complex

A.1, 2 and 3 are all correct statements about such a wave
B.2 and 3 only are correct
C.1 and 3 only are correct
D.1 and 2 only are correct

Answer: D

Correct answer: 1 and 2 only are correct

Statements 1 and 2 are correct: in a lossless medium the electric and magnetic fields are exactly in phase, and both lie transverse to the direction of travel. Statement 3 is wrong - \( \eta = \sqrt{\mu/\epsilon} \) is purely real when there is no loss, and it is exactly that reality which puts the two fields in phase. A complex impedance is the signature of a lossy medium.

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49.

Consider the following statements about reflection at normal incidence.

1. Equal impedances give no reflection 2. A perfect conductor gives a coefficient of minus one 3. The transmitted amplitude is one minus the reflection coefficient

A.1 and 2 only are correct
B.1, 2 and 3 are all correct statements about normal incidence
C.2 and 3 only are correct
D.1 and 3 only are correct

Answer: A

Correct answer: 1 and 2 only are correct

Statements 1 and 2 are correct: matched impedances give no reflection, and a perfect conductor gives \( \Gamma = -1 \). Statement 3 has the sign wrong - the transmission coefficient is \[ \tau = 1 + \Gamma, \] not \( 1-\Gamma \). The relation looks odd until one notices that the tangential electric field must be continuous, so the transmitted amplitude equals the incident plus the reflected.

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50.

Which of the following does NOT change when a wave passes from free space into a dielectric?

A.the intrinsic impedance
B.the frequency
C.the wavelength
D.the phase velocity

Answer: B

The frequency is fixed by the source and is the same everywhere along the path, which is why the wavelength must change when the velocity does. The other three all scale with the square root of the relative permittivity, two downward and one downward as well.

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