/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 A circular wire loop has a radiu... [FREE SOLUTION] | 91影视

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Chapter 32: Problem 42

A circular wire loop has a radius of \(7.50 \mathrm{~cm}\). A sinusoidal electromagnetic plane wave traveling in air passes through the loop, with the direction of the magnetic field of the wave perpendicular to the plane of the loop. The intensity of the wave at the location of the loop is \(0.0275 \mathrm{~W} / \mathrm{m}^{2}\), and the wavelength of the wave is \(6.90 \mathrm{~m}\). What is the maximum emf induced in the loop?

Short Answer

Expert verified
The maximum induced emf is approximately 0.0224 V.

Step by step solution

01

Determine Magnetic Field Amplitude

To find the magnetic field amplitude \( B_0 \), use the formula: \[ I = \frac{1}{2} c \varepsilon_0 E_0^2 = \frac{1}{2} \frac{cB_0^2}{\mu_0} \] Given the intensity \( I = 0.0275\, \text{W/m}^2 \), calculate \( B_0 \): \[ B_0 = \sqrt{\frac{2I\mu_0}{c}} \] where \( c = 3 \times 10^8 \, \text{m/s} \) and \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \). \[ B_0 = \sqrt{\frac{2 \times 0.0275 \times 4\pi \times 10^{-7}}{3 \times 10^8}} \approx 4.62 \times 10^{-9} \, \text{T} \]
02

Calculate Angular Frequency

Find the angular frequency \( \omega \) using the wave speed \( c \) and wavelength \( \lambda \): \[ \omega = \frac{2\pi c}{\lambda} \] \[ \omega = \frac{2\pi \times 3 \times 10^8}{6.90} \approx 2.73 \times 10^8 \, \text{rad/s} \]
03

Compute Induced EMF

The induced emf can be calculated using Faraday's law of electromagnetic induction: \[ \text{emf}_{\text{max}} = A B_0 \omega \] where \( A = \pi r^2 \) is the area of the loop. With the radius \( r = 0.075 \, \text{m} \), \[ A = \pi (0.075)^2 \approx 0.0177 \, \text{m}^2 \]. Then, compute the emf: \[ \text{emf}_{\text{max}} = 0.0177 \times 4.62 \times 10^{-9} \times 2.73 \times 10^8 \approx 2.24 \times 10^{-2} \, \text{V} \]

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Key Concepts

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

Maxwell's Equations
Maxwell's Equations are fundamental to understanding electromagnetism. They describe how electric and magnetic fields interact and how they are influenced by charges and currents. There are four key equations, each with its own significance:
  • Gauss's Law for Electricity: This relates the electric field to the charge distribution. It states that the electric flux through a closed surface is proportional to the charge enclosed.
  • Gauss's Law for Magnetism: It indicates that magnetic field lines are closed loops, implying no magnetic monopoles exist. The magnetic flux through a closed surface is zero.
  • Faraday's Law of Induction: This explains how a changing magnetic field can induce an electric field. It is crucial in understanding electromagnetic induction.
  • Ampere-Maxwell Law: This law relates magnetic fields to the electric current and a changing electric field, introducing the concept of displacement current.
Each of these laws plays a role in deriving the behavior of electromagnetic waves and their interplay with matter. By integrating them, Maxwell established that light is an electromagnetic wave, a breakthrough in physics.
Faraday's Law
Faraday's Law of Electromagnetic Induction is a cornerstone of electromagnetism. It states that a change in the magnetic environment of a coil of wire will induce an electromotive force (emf) in the coil. The induced emf can generate current if the circuit is closed. The key part of Faraday's law is that it depends on how quickly the magnetic field changes:
  • When the magnetic field inside a coil increases or decreases, it induces a voltage.
  • The faster the change, the greater the induced emf.
  • The direction of the induced current is such that it opposes the change in magnetic flux, described by Lenz's Law.
Mathematically, Faraday's law is expressed as: \[ \text{emf} = - \frac{d\Phi_B}{dt} \]where \( \Phi_B \) is the magnetic flux. In the exercise, we use this principle to calculate the maximum induced emf in a wire loop due to an incoming electromagnetic wave. The negative sign indicates the direction of the induced emf opposes the change, embodying Lenz's Law.
Electromagnetic Waves
Electromagnetic waves are created by oscillating electric and magnetic fields. These fields propagate through space and can transport energy, as demonstrated in light radio waves and X-rays.
  • Electromagnetic waves travel at the speed of light, approximately \(3 \times 10^8 \text{ m/s}\) in a vacuum.
  • They are transverse waves, meaning the oscillations are perpendicular to the direction of wave travel.
  • These waves do not require a medium to travel through, allowing them to move through the vacuum of space.
Maxwell's findings that light itself is an electromagnetic wave unified the electric and magnetic fields concepts. In the given exercise, a sinusoidal electromagnetic wave is considered traveling through a loop, allowing us to explore how electromagnetic induction can occur via this natural phenomenon.Understanding the behavior of electromagnetic waves is crucial for fields such as communication technology, radar, and even studying light's properties as waves and particles.

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Most popular questions from this chapter

Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of 10\(^{12}\) W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0 \(\mu\)m in diameter, with the pulse lasting for 4.0 ns with an average power of 2.0 \(\times\) 10\(^{12}\) W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in W/m\(^2\)) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

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 electricfield vector \(\vec{E}\) 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 \(\vec{B}\) at the same point? (c) What are the magnitude and direction of the Poynting vector \(\vec{S}\) at the same point? (The direction of \(\vec{S}\) 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.

A monochromatic light source with power output 60.0 W radiates light of wavelength 700 nm uniformly in all directions. Calculate \(E_{max}\) and \(B_{max}\) for the 700-nm light at a distance of 5.00 m from the source.

For a sinusoidal electromagnetic wave in vacuum, such as that described by Eq. (32.16), show that the \(average\) energy density in the electric field is the same as that in the magnetic field.

A satellite 575 km above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 MHz uniformly in all directions, with a power of 25.0 kW. (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0 cm by 40.0 cm oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?
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