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Chapter 21: Problem 51

An electromagnetic wave in free space has an electric field of amplitude \(330 \mathrm{~V} / \mathrm{m}\). Find the amplitude of the corresponding magnetic field.

Short Answer

Expert verified
The amplitude of the corresponding magnetic field is \(1.1 x 10^{-6} T\).

Step by step solution

01

Write Down Given Values and the Formula

The electric field, E is given as 330 V/m. The speed of light, c, is known to be approximately \(3 \times 10^8 m/s\). The formula to use is B = E/c.
02

Insert Values into Formula

Substitute E = 330 V/m and c = \(3 \times 10^8 m/s\) into the formula: B = \(330/(3 x 10^{8}) = 1.1 x 10^{-6} T\).
03

Evaluate the Expression

Calculate the equation to find the corresponding magnetic field amplitude, B = \(1.1 x 10^{-6} T\).

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Key Concepts

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

Electric Field
The electric field is a fundamental component of electromagnetic waves. It describes the force that a charge would experience in the vicinity of another charge or electric source. In simpler terms, an electric field is generated by electric charges and can affect other charges around it.

An electric field is typically represented by the symbol \(E\) and is measured in volts per meter (V/m). For the electromagnetic waves propagating through free space, the electric field oscillates perpendicular to the direction of wave propagation.
  • The strength of the electric field defines the amplitude of the wave.
  • The given problem specifies an electric field amplitude of 330 V/m.
  • This value portrays how much energy is carried by the electric field of the wave.
Studying electric fields helps grasp how electromagnetic waves move energy across space.
Magnetic Field
The magnetic field is another vital part of an electromagnetic wave, closely interlinked with the electric field. It arises from moving electric charges or acts around magnets. For electromagnetic waves, the magnetic field oscillates at a right angle to both the electric field and the direction of wave travel.

The magnetic field is denoted by \(B\) and measured in teslas (T). In an electromagnetic wave, the electric and magnetic fields sustain each other in equilibrium.
  • In this scenario, we are tasked with finding the amplitude of the magnetic field, which can be derived using the formula \(B = \frac{E}{c}\).
  • By inserting the electric field amplitude of 330 V/m and the speed of light, we find \(B = 1.1 \times 10^{-6} T\).
  • The result tells us about the intensity of the magnetic field for the given wave.
Understanding magnetic fields in the context of electromagnetic waves aids in exploring phenomena like electromagnetic induction and communication signals.
Speed of Light
The speed of light, typically denoted as \(c\), is a universal constant that represents how fast light travels in a vacuum. Its approximate value is \(3 \times 10^8\ m/s\), playing a crucial role in the behavior of electromagnetic waves.

Electromagnetic waves, including visible light, radio waves, and microwaves, all travel at this speed in a vacuum, setting the maximum velocity in space.
  • The speed of light is instrumental in determining the relationship between the electric and magnetic fields in electromagnetic waves.
  • By using the equation \(B = \frac{E}{c}\), we relate the electric field to the magnetic field, utilizing the speed of light as a proportional constant.
  • This relationship highlights how changes in one field necessitate changes in the other to maintain the wave's properties.
Comprehending the speed of light illuminates fundamental physical laws and interactions that define our universe.

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Most popular questions from this chapter

The U.S. Navy has long proposed the construction of extremely low frequency (ELF waves) communications systems; such waves could penetrate the oceans to reach distant submarines. Calculate the length of a quarterwavelength antenna for a transmitter generating ELF waves of frequency \(75 \mathrm{~Hz}\). How practical is this antenna?

The output voltage of an AC generator is given by \(\Delta v=(170 \mathrm{~V}) \sin (60 \pi t)\). The generator is connected across a \(20.0-\Omega\) resistor. By inspection, what are the (a) maximum voltage and (b) frequency? Find the (c) \(\mathrm{rms}\) voltage across the resistor, (d) rms current in the resistor, (e) maximum current in the resistor, and (f) power delivered to the resistor. \((g)\) Should the argument of the sine function be in degrees or radians? Compute the current when \(t=0.0050 \mathrm{~s}\).

A step-down transformer is used for recharging the batteries of portable devices. The turns ratio \(N_{2} / N_{1}\) for a particular transformer used in a CD player is 1:13. When used with \(120-\mathrm{V}\) (rms) household service, the transformer draws an rms current of \(250 \mathrm{~mA}\). Find the (a) rms output voltage of the transformer and (b) power delivered to the CD player.

A \(200-\Omega\) resistor is connected in series with a \(5.0-\mu \mathrm{F}\) capacitor and a \(60-\mathrm{Hz}, 120-\mathrm{Vrms}\) line. If electrical energy costs \(\$ 0.080 / \mathrm{kWh}\), how much does it cost to leave this circuit connected for \(24 \mathrm{~h}\) ?

A resistor \(\left(R=9.00 \times 10^{2} \Omega\right)\), a capacitor \((C=0.250 \mu \mathrm{F})\), and an inductor \((L=2.50 \mathrm{H})\) are connected in series across a \(2.40 \times 10^{2}-\mathrm{Hz}\) AC source for which \(\Delta V_{\max }=\) \(1.40 \times 10^{2} \mathrm{~V}\), Calculate (a) the impedance of the circuit, (b) the maximum current delivered by the source, and (c) the phase angle between the current and voltage. (d) Is the current leading or lagging the voltage?
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