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Chapter 34: Problem 3

The speed of an electromagnetic wave traveling in a transparent nonmagnetic substance is \(v=1 / \sqrt{\kappa \mu_{0} \epsilon_{0}},\) where \(\kappa\) is the dielectric constant of the substance. Determine the speed of light in water, which has a dielectric constant at optical frequencies of 1.78.

Short Answer

Expert verified
The speed of light in water, calculated using the formula is approximately \(2.25 \times 10^8 m/s\).

Step by step solution

01

Recognize the Given Information

From the statement of the problem, we have dielectric constant \(\kappa\) = 1.78 for water. The constants \(\mu_{0}\) and \(\epsilon_{0}\) are respectively \(4\pi \times 10^{-7} T \cdot m/A\) and \(8.85 \times 10^{-12} C^2/(N \cdot m^2)\)
02

Substitute the Given Information into the Formula

Substitute \(\kappa\), \(\mu_{0}\), and \(\epsilon_{0}\) into the speed of electromagnetic wave formula. That is, \(v = 1 / \sqrt{\kappa \mu_{0} \epsilon_{0}}\)
03

Perform the Calculation

Evaluate the expression to find the speed of light in water. Remember that since all the values are properly substituted in SI units, the speed obtained will also be in SI unit 'm/s'.

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Key Concepts

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

Electromagnetic Waves
Electromagnetic waves are celestial travelers gracing us with their presence, carrying energy without needing a medium. Imagine them moving like ripples in a pond, but with electrifying differences. Unlike sound waves, they can move through the vacuum of space. This adaptability is why we can experience light from the sun and communicate over vast distances with radio and television signals.

These waves, including visible light, X-rays, and microwaves, share a common characteristic: they travel at the speed of light, approximately 300,000 kilometers per second in a vacuum. However, their speed can vary depending on the medium, such as water or glass, through which they pass. The concept of electromagnetic waves is central to understanding phenomena like reflection, refraction, and wave interference.
  • Electromagnetic waves are crucial for various technologies, such as communication and medical imaging.
  • They are characterized by frequencies and wavelengths, dictating their specific type within the electromagnetic spectrum.
  • Visible light is only a small part of this spectrum, spanning from red to violet.
Understanding electromagnetic waves aids in comprehending how information travels and why light behaves differently in various substances.
Dielectric Constant
The dielectric constant, or relative permittivity, is a measure of a material's tendency to store electrical energy in an electric field. Think of it as an indicator of how much a substance can "squeeze" the electric field within it. This constant is particularly important for materials like water, glass, and ceramics, as it affects how they influence electromagnetic waves passing through them.

In mathematical terms, the dielectric constant \( \kappa \) modifies the speed of electromagnetic waves in a medium, in comparison to their speed in a vacuum. The formula used in calculating the speed of light in a substance, such as water, includes this constant, reflecting its vital role.
  • The dielectric constant is a dimensionless number.
  • It helps in determining the capacitance of capacitors.
  • A higher dielectric constant indicates a better capability to reduce the speed of electromagnetic waves in the medium.
When calculating the speed of light in various substances, understanding their dielectric constants can offer insights into optical properties and conductivity.
Nonmagnetic Substances
Nonmagnetic substances are materials that do not exhibit magnetism, meaning they don't react significantly to magnetic fields. Common materials like wood, plastic, and water fall into this category. Despite this, they can still interact with electromagnetic waves through their dielectric properties.

In the context of this exercise, considering water as a nonmagnetic substance is crucial. While water molecules are polar, showing some magnetic characteristics due to the arrangement of their atoms, water is generally considered nonmagnetic for calculations involving the propagation of electromagnetic waves.
  • Nonmagnetic substances are significant when exploring electrical insulation and wave propagation.
  • These materials help in focusing on electric aspects without magnetic interference.
  • They contribute to practical applications, like in electronics, where avoiding magnetic influence is key.
The understanding of nonmagnetic substances becomes relevant when analyzing wave behaviors purely through their electric characteristics, without accounting for magnetic interactions.

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Most popular questions from this chapter

A plane electromagnetic wave varies sinusoidally at \(90.0 \mathrm{MHz}\) as it travels along the \(+x\) direction. The peak value of the electric field is \(2.00 \mathrm{mV} / \mathrm{m},\) and it is directed along the \(\pm y\) direction. (a) Find the wavelength, the period, and the maximum value of the magnetic field. (b) Write expressions in SI units for the space and time variations of the electric field and of the magnetic field. Include numerical values and include subscripts to indicate coordinate directions. (c) Find the average power per unit area that this wave carries through space. (d) Find the average energy density in the radiation (in joules per cubic meter). (e) What radiation pressure would this wave exert upon a perfectly reflecting surface at normal incidence?

In the absence of cable input or a satellite dish, a television set can use a dipole-receiving antenna for VHF channels and a loop antenna for UHF channels (Fig. Q34.12). The UHF antenna produces an emf from the changing magnetic flux through the loop. The TV station broadcasts a signal with a frequency \(f\), and the signal has an electric-field amplitude \(E_{\max }\) and a magnetic-field amplitude \(B_{\max }\) at the location of the receiving antenna. (a) Using Faraday's law, derive an expression for the amplitude of the emf that appears in a single-turn circular loop antenna with a radius \(r,\) which is small compared with the wavelength of the wave. (b) If the electric field in the signal points vertically, what orientation of the loop gives the best reception?

Figure 34.10 shows a Hertz antenna (also known as a halfwave antenna, because its length is \(\lambda / 2\) ). The antenna is located far enough from the ground that reflections do not significantly affect its radiation pattern. Most AM radio stations, however, use a Marconi antenna, which consists of the top half of a Hertz antenna. The lower end of this (quarter-wave) antenna is connected to Earth ground, and the ground itself serves as the missing lower half. What are the heights of the Marconi antennas for radio stations broadcasting at (a) \(560 \mathrm{kHz}\) and (b) \(1600 \mathrm{kHz}^{2}\)

A 1.00 -m-diameter mirror focuses the Sun's rays onto an absorbing plate \(2.00 \mathrm{cm}\) in radius, which holds a can containing \(1.00 \mathrm{L}\) of water at \(20.0^{\circ} \mathrm{C}\) (a) If the solar intensity is \(1.00 \mathrm{kW} / \mathrm{m}^{2},\) what is the intensity on the absorbing plate? (b) What are the maximum magnitudes of the fields \(\mathbf{E}\) and \(\mathbf{B} ?\) (c) If \(40.0 \%\) of the energy is absorbed, how long does it take to bring the water to its boiling point?

Consider a small, spherical particle of radius \(r\) located in space a distance \(R\) from the Sun. (a) Show that the ratio \(F_{\mathrm{rad}} / F_{\mathrm{grav}}\) is proportional to \(1 / r,\) where \(F_{\mathrm{rad}}\) is the force exerted by solar radiation and \(F_{\mathrm{grav}}\) is the force of gravitational attraction. (b) The result of part (a) means that, for a sufficiently small value of \(r,\) the force exerted on the particle by solar radiation exceeds the force of gravitational attraction. Calculate the value of \(r\) for which the particle is in equilibrium under the two forces. (Assume that the particle has a perfectly absorbing surface and a mass density of \(1.50 \mathrm{g} / \mathrm{cm}^{3} .\) Let the particle be located \(3.75 \times 10^{11} \mathrm{m}\) from the Sun, and use \(214 \mathrm{W} / \mathrm{m}^{2}\) as the value of the solar intensity at that point.)
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