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Chapter 31: Problem 10

A dipole antenna is located at the origin with its axis along the \(z\) -axis. As electric current oscillates up and down the antenna, polarized electromagnetic radiation travels away from the antenna along the positive \(y\) -axis. What are the possible directions of electric and magnetic fields at point \(A\) on the \(y\) -axis? Explain.

Short Answer

Expert verified
Answer: At point A on the y-axis, the possible directions of the electric field are parallel to the z-axis, and the possible directions of the magnetic field are parallel to the x-axis.

Step by step solution

01

Understand dipole antenna radiation mechanism

The dipole antenna has its axis along the z-axis, and current oscillates up and down along this axis, producing electromagnetic radiation. The radiation is polarized, and it is emitted along the positive y-axis.
02

Identify coordinates and directions

We need to find the direction of the electric and magnetic fields at point A, which is on the y-axis. The y-axis is the direction of propagation, meaning that the radiation is emitted along this axis. Since the electric field is always perpendicular to the magnetic field, and the fields must also be perpendicular to the direction of propagation (y-axis), these fields must lie on the xz-plane.
03

Determine electric field direction

At point A on the y-axis, since the electric field must be perpendicular to the y-axis, the possible directions of the electric field should be in the xz-plane. Because the antenna's axis is along the z-axis and electric current oscillates along this axis, EM radiation is polarized in the z-axis, making the electric field parallel to the z-axis at point A. The electric field direction at point A is parallel to the z-axis.
04

Determine magnetic field direction

To find the direction of the magnetic field at point A, we can use the property that the magnetic field is always perpendicular to the electric field and the direction of propagation. Since the electric field lies along the z-axis, and the direction of propagation is along the y-axis, the magnetic field must be along the x-axis. The magnetic field direction at point A is parallel to the x-axis. In conclusion, at point A on the y-axis, the possible directions of the electric field are parallel to the z-axis, and the possible directions of the magnetic field are parallel to the x-axis.

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Most popular questions from this chapter

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy \(E=\hbar \omega,\) where \(\hbar=1.054573 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}\) is Planck's reduced constant and \(\omega\) is the angular frequency of the wave. a) Find the momentum of a photon. b) Find the angular momentum of a photon. Photons are circularly polarized; that is, they are described by a superposition of two plane-polarized waves with equal field amplitudes, equal frequencies, and perpendicular polarizations, one-quarter of a cycle \(\left(90^{\circ}\right.\) or \(\pi / 2\) rad \()\) out of phase, so the electric and magnetic field vectors at any fixed point rotate in a circle with the angular frequency of the waves. It can be shown that a circularly polarized wave of energy \(U\) and angular frequency \(\omega\) has an angular momentum of magnitude \(L=U / \omega .\) (The direction of the angular momentum is given by the thumb of the right hand, when the fingers are curled in the direction in which the field vectors circulate. c) The ratio of the angular momentum of a particle to \(\hbar\) is its spin quantum number. Determine the spin quantum number of the photon.

A tiny particle of density \(2000 . \mathrm{kg} / \mathrm{m}^{3}\) is at the same distance from the Sun as the Earth is \(\left(1.50 \cdot 10^{11} \mathrm{~m}\right)\). Assume that the particle is spherical and perfectly reflecting. What would its radius have to be for the outward radiation pressure on it to be \(1.00 \%\) of the inward gravitational attraction of the Sun? (Take the Sun's mass to be \(\left.2.00 \cdot 10^{30} \mathrm{~kg} .\right)\)

Three FM radio stations covering the same geographical area broadcast at frequencies \(91.1,91.3,\) and \(91.5 \mathrm{MHz},\) respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5? Use \(c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\), and calculate the wavelength to an uncertainty of \(1 \mathrm{~mm} .\)

A laser beam takes 50.0 ms to be reflected back from a totally reflecting sail on a spacecraft. How far away is the sail?

Calculate the average value of the Poynting vector, \(S_{\text {ave }}\) for an electromagnetic wave having an electric field of amplitude \(100 . \mathrm{V} / \mathrm{m}\) a) What is the average energy density of this wave? b) How large is the amplitude of the magnetic field?
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