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Chapter 32: Problem 34

An electromagnetic standing wave in air has frequency 75.0\(\mathrm { MHz }\) . (a) What is the distance between nodal planes of the \(\vec { \boldsymbol { E } }\) field? (b) What is the distance between a nodal plane of \(\vec { \boldsymbol { E } }\) and the closest nodal plane of \(\vec { \boldsymbol { B } }\) ?

Short Answer

Expert verified
(a) 2.00 m; (b) 1.00 m.

Step by step solution

01

Determine the Wavelength

The speed of electromagnetic waves in air is approximately the speed of light, which is \( c = 3.00 \times 10^8 \text{ m/s} \). The wavelength \( \lambda \) of the wave can be found using the formula \( \lambda = \frac{c}{f} \), where \( f = 75.0 \text{ MHz} = 75.0 \times 10^6 \text{ Hz} \). Calculate \( \lambda \): \[ \lambda = \frac{3.00 \times 10^8}{75.0 \times 10^6} = 4.00 \text{ m} \]
02

Find the Distance Between Nodal Planes of the E Field

In a standing wave, the nodal planes of the electric field (\( \vec{E} \)) are spaced half a wavelength apart. Therefore, the distance between nodal planes is \( \frac{\lambda}{2} \). Using the wavelength from Step 1:\[ \frac{\lambda}{2} = \frac{4.00}{2} = 2.00 \text{ m} \]
03

Calculate the Distance Between a Nodal Plane of E and B Fields

The nodal planes of the electric field (\( \vec{E} \)) and the magnetic field (\( \vec{B} \)) are staggered by a quarter wavelength. This means the distance between a nodal plane of \( \vec{E} \) and the nearest nodal plane of \( \vec{B} \) is \( \frac{\lambda}{4} \). Using \( \lambda = 4.00 \text{ m} \):\[ \frac{\lambda}{4} = \frac{4.00}{4} = 1.00 \text{ m} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nodal Planes in Electromagnetic Standing Waves
Nodal planes are essential components of any standing wave, as they are the locations where certain wave characteristics, like amplitude, go to zero. When discussing electromagnetic standing waves, which consist of electric (\(\vec{E}\)) and magnetic (\(\vec{B}\)) fields, these nodal planes become particularly crucial. In an electromagnetic standing wave, the electric field nodes and the magnetic field nodes are not at the same positions, leading to unique spatial relationships. In the context of determining distances between nodal planes, understanding the nature of these fields becomes useful. Every nodal plane for a standing wave is aligned such that there's no oscillation at that point鈥攊t's like the wave "pauses" there.
  • Electric field nodal planes occur at zero amplitude of electric oscillation.
  • Magnetic field nodal planes occur at zero amplitude of magnetic oscillation.
Understanding the Electric Field
The electric field (\(\vec{E}\)) in an electromagnetic wave reflects how electric charges are influenced by the wave. For a standing wave, the electric field is structured so that it stays 'stationary' in terms of its oscillations. This means parts of the wave have high electric field intensity while other parts, namely the nodal planes, have none.In standing waves, these nodal planes for the electric field appear at points spaced out by half a wavelength (\(\frac{\lambda}{2}\)). This spacing is a direct result of the wave reflecting back on itself and the interference patterns that form.
  • A wave's electric field affects charged particles within the field's path.
  • The field's amplitude is an indicator of the wave's strength in any particular spot.
Characteristics of the Magnetic Field
Alongside the electric field, the magnetic field (\(\vec{B}\)) makes up the second half of the electromagnetic wave spectrum. Much like the electric field, the magnetic field has its own nodal planes. These are regions where the magnetic influence of the wave does not manifest.By understanding the interplay between electric and magnetic nodal planes, one can discern the complete behavior of a standing electromagnetic wave. In a given row of nodal planes, the magnetic nodal planes intersperse with those of the electric field. They are spaced out by a quarter wavelength (\(\frac{\lambda}{4}\)) from one another. This staggering provides a unique positional relationship that is crucial for analyzing wave behavior.
  • The magnetic field applies a force on moving charges perpendicular to their velocity.
  • Its nodal planes are separated in distance from those of the electric field.
The Role of Wavelength in Standing Waves
Wavelength is fundamental when discussing wave properties, as it defines the spatial periodicity of the wave itself. In an electromagnetic standing wave, the wavelength (\(\lambda\)) is the distance over which the wave's shape repeats.Understanding how wavelength relates to nodal planes is crucial to mastering the concept of standing waves. The distance between nodal planes for the electric field within a standing wave is exactly half of the wavelength. This measurement is critical when trying to determine properties such as the speed of the wave or its frequency given a nodal pattern.
  • Wavelength inversely affects frequency of the wave.
  • The spacing between nodes is derived directly from the wave's wavelength.
Frequency and Its Impact on Waves
Frequency, a critical wave parameter, tells us how many wave cycles occur in a given time period. For electromagnetic waves, this is measured in hertz (Hz), and has significant impacts on how these waves behave.With a higher frequency, the standing wave's characteristics, such as its wavelength, are affected given that wavelength is inversely proportional to frequency: \(\lambda = \frac{c}{f}\), where \(c\) is the speed of light. In the given exercise, where a frequency of 75 MHz was utilized, one observed how this frequency dictated not only the wavelength but also the formation of nodal planes.
  • Higher frequency equals shorter wavelength and more nodal planes.
  • Frequency ultimately defines the wave's energy and movement character.

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Most popular questions from this chapter

Laser Surgery. Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina weld the detached portions back into place. In one such procedure, a laser beam has a wavelength of 810\(\mathrm { nm }\) and delivers 250\(\mathrm { mW }\) of power spread over a circular spot 510\(\mu \mathrm { m }\) in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of 1.34 (a) If the laser pulses are each 1.50 ms long, how much energy is delivered to the retina with each pulse? (b) What average pressure does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?

An electromagnetic standing wave in a certain material has frequency \(1.20 \times 10 ^ { 10 } \mathrm { Hz }\) and speed of propagation \(2.10 \times\) \(10 ^ { 8 } \mathrm { m } / \mathrm { s } .\) (a) What is the distance between a nodal plane of \(\vec { \boldsymbol { B } }\) and the closest antinodal plane of \(\vec { \boldsymbol { B } } ?\) (b) What is the distance between an antinodal plane of \(\vec { \boldsymbol { E } }\) and the closest antinodal plane of \(\vec { \boldsymbol { B } }\) ? (c) What is the distance between a nodal plane of \(\vec { \boldsymbol { E } }\) and the closest nodal plane of \(\vec { \boldsymbol { B } }\) ?

Television Broadcasting. Public television station KQED in San Francisco broadcasts a sinusoidal radio signal at a power of 316\(\mathrm { kW }\) . Assume that the wave spreads out uniformly into a hemisphere above the ground. At a home 5.00\(\mathrm { km }\) away from the antenna, (a) what average pressure does this wave exert on a totally reflecting surface, (b) what are the amplitudes of the electric and magnetic fields of the wave, and (c) what is the average density of the energy this wave carries? (d) For the energy density in part (c), what percentage is due to the electric field and what percentage is due to the magnetic field?

Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) \(\vec { \boldsymbol { E } }\) in the \(+ x\) -direction, \(\vec { \boldsymbol { B } }\) in the \(+ y\) - direction; (b) \(\vec { \boldsymbol { E } }\) in the \(- y\) - direction, \(\vec { \boldsymbol { B } }\) in the \(+ x\) -direction; \(( \mathrm { c } ) \vec { \boldsymbol { E } }\) in the \(+ z\) -direction, \(\vec { \boldsymbol { B } }\) in the \(- x\) -direction; (d) \(\vec { \boldsymbol { E } }\) in the \(+ y\) -direction, \(\vec { \boldsymbol { B } }\) in the \(- z\) -direction.

A cylindrical conductor with a circular cross section has a radius \(a\) and a resistivity \(\rho\) and carries a constant current \(I\) (a) What are the magnitude and direction of the electric-field vector \(E\) at a point just inside the wire at a distance \(a\) from the axis? (b) What are the magnitude and direction of the magneticfield vector \(\vec { \boldsymbol { B } }\) at the same point? \(( \mathrm { c } )\) What are the magnitude and direction of the Poynting vector \(\vec { \boldsymbol { S } }\) at the same point? (The direction of \(\vec { \boldsymbol { S } }\) is the direction in which electromagnetic energy flows into or out of the conductor.) (d) Use the result in part (c) to find the rate of flow of energy into the volume occupied by a length \(l\) of the conductor. (Hint: Integrate \(\vec { S }\) over the surface of this volume.) Compare your result to the rate of generation of thermal energy in the same volume. Discuss why the energy dissipated in a current carrying conductor, due to its resistance, can be thought of as entering through the cylindrical sides of the conductor.
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