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Chapter 8: Problem 2

A charged particle in an electromagnetic field experiences the Lorentz force $$ \mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}), $$ where \(q\) is the charge and \(\mathbf{v}\) the (vector) velocity of the particle. Show that an electromagnetic wave in free space acts on a charged particle primarily through its electric field, the magnetic interaction being smaller by at least the ratio \(|\mathbf{v}| / c\).

Short Answer

Expert verified
As an electromagnetic wave travels through free space, explain whether the magnetic field or electric field has a more significant effect on a charged particle and why. The electric field has a more significant effect on a charged particle when an electromagnetic wave travels through free space. This is due to the electric and magnetic forces acting on the charged particle. When comparing the magnitudes of these forces (electric force, \(\mathbf{F_E}\), and magnetic force, \(\mathbf{F_B}\)), the magnetic interaction is smaller than the electric interaction by a factor of the ratio \(|\mathbf{v}|/c\), where \(\mathbf{v}\) is the particle's velocity and \(c\) is the speed of light. This means that the charged particle experiences greater acceleration and force from the electric field compared to the magnetic field in free space.

Step by step solution

01

Calculate the Magnitudes of Electric and Magnetic Forces

First, we need to find the magnitudes of electric force \(\mathbf{F_E}\) and magnetic force \(\mathbf{F_B}\) experienced by the charged particle. From the Lorentz force equation, we know that the electric force and magnetic forces are given by: $$ \mathbf{F_E} = q\mathbf{E}, $$ and $$ \mathbf{F_B} = q(\mathbf{v} \times \mathbf{B}). $$
02

Calculate the Ratio of Electric and Magnetic Fields

We know that in an electromagnetic wave in free space, the magnitudes of electric field \(E\) and magnetic field \(B\) are related by the speed of light \(c\), which can be given by: $$ c = \frac{E}{B}. $$
03

Express \(\mathbf{F_B}\) in terms of Electric Field, Speed of Light and Charged Particle Velocity

Since we are interested in the comparison between \(\mathbf{F_E}\) and \(\mathbf{F_B}\), let's express the magnitude of the magnetic force in terms of the electric field. Using \(E = cB\), we can write: $$ \mathbf{F_B} = q(\mathbf{v} \times \frac{1}{c}\mathbf{E}), $$ which simplifies to $$ \mathbf{F_B} = \frac{q}{c}(\mathbf{v} \times \mathbf{E}). $$
04

Compare Magnitudes of \(\mathbf{F_E}\) and \(\mathbf{F_B}\)

Now let's compare the magnitudes of the electric and magnetic forces acting on the charged particle: $$ \frac{|\mathbf{F_B}|}{|\mathbf{F_E}|} = \frac{|\frac{q}{c}(\mathbf{v} \times \mathbf{E})|}{|q\mathbf{E}|}. $$ Notice that \(q\) cancels out on both sides, so we get: $$ \frac{|\mathbf{F_B}|}{|\mathbf{F_E}|} = \frac{|\mathbf{v} \times \mathbf{E}|}{c|\mathbf{E}|}. $$
05

Estimate the Smallest Value of the Ratio

Let's find the smallest value of this ratio. We know that the magnitude of the cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(|\mathbf{a}||\mathbf{b}|\sin\theta\), where \(\theta\) is the angle between the vectors. In this case, \(\mathbf{a} = \mathbf{v}\) and \(\mathbf{b} = \mathbf{E}\), hence $$ \frac{|\mathbf{F_B}|}{|\mathbf{F_E}|} = \frac{|\mathbf{v}\sin\theta|}{c}. $$ The maximum value of \(\sin\theta\) is 1, so the smallest value of the above ratio is $$ \frac{|\mathbf{F_B}|}{|\mathbf{F_E}|} \ge \frac{|\mathbf{v}|}{c}. $$ Therefore, the magnetic interaction is smaller than the electric interaction by at least the ratio \(|\mathbf{v}|/c\).

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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 waves that are composed of oscillating electric and magnetic fields, which propagate through space at the speed of light. These waves are fundamental to many aspects of physics and everyday technology, including light, radio waves, and X-rays.
  • Propagation: Electromagnetic waves travel through a vacuum without the need for a medium, which is unique compared to mechanical waves that require a medium like water or air.
  • Wave Nature: They have a wave-like nature, characterized by their wavelength and frequency, which determine the type of electromagnetic radiation (such as visible light or microwaves).
When an electromagnetic wave encounters a charged particle, it can exert a force on it. This interaction is crucial in understanding how light and other electromagnetic waves influence matter.
Electric Field
An electric field is a region around a charged particle where other charged particles experience a force. The strength of this force depends on the amount of charge and the distance from the charge producing the field.
  • Vector Quantity: Electric fields have both magnitude and direction, represented as vectors.
  • Source of Force: In the context of the Lorentz force, the electric field is crucial because it exerts a force directly proportional to the charge of the particle.
  • Interaction with Particles: Charged particles within an electric field experience a force given by \[\mathbf{F_E} = q\mathbf{E}\] where \(q\) is the charge and \(\mathbf{E}\) is the electric field vector.
Electric fields are integral in understanding how charges interact and how electromagnetic waves influence particles. They play a significant role in the force exerted by an electromagnetic wave on a charged particle, as shown by the Lorentz force equation.
Magnetic Field
Magnetic fields are created by moving electric charges and can influence other charges within their reach. These fields are integral to the concept of electromagnetic waves, as they work together with electric fields to propagate the wave.
  • Vector Field: Like electric fields, magnetic fields are also vector fields, having both direction and magnitude.
  • Part of Lorentz Force: When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field, given by \[\mathbf{F_B} = q(\mathbf{v} \times \mathbf{B})\].
  • Weaker Influence: In electromagnetic waves, the magnetic field influences charged particles but typically much less than the electric field. This is because the force from the magnetic field involves the cross product of velocity and magnetic field, which depends on the particle's speed and the angle between velocity and the magnetic field.
Understanding magnetic fields helps in assessing their contributions to the forces acting on charged particles in electromagnetic fields.
Particle Charge
Particle charge refers to the intrinsic property of particles that causes them to experience attractive or repulsive forces within an electric or magnetic field. Charge is a fundamental characteristic that defines how particles interact.
  • Quantization: Charge comes in discrete amounts and is measured in Coulombs. Typically, charges are either positive or negative.
  • Influence on Forces: The charge of a particle is crucial when calculating the Lorentz force experienced by a particle in an electromagnetic field.
  • Proportional to Forces: The magnitude of both the electric and magnetic forces acting on a charged particle is directly proportional to the amount of charge, as seen in \(\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\).
Understanding the nature of charge allows us to predict and calculate the behavior of particles under the influence of electromagnetic fields, providing a solid foundation for exploring more complex electromagnetic interactions.

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Most popular questions from this chapter

For the TE \(_{10}\) mode in rectangular waveguide, find the values of \(x\) at which the magnetic field is circularly polarised; i.e., the \(B_{x}\) and \(B\), components are equal in magnitude and \(90^{\circ}\) out of phase in time. (This feature is exploited in some waveguide devices known as directional couplers and isolators) Anstoer: \(\sin (\pi x / a)=\lambda_{0} / 2 a\).

From \((8.7 .18)\) show that the phase velocity of the wave in a waveguide is $$ c_{p}=\frac{\omega}{\kappa_{x}}=\frac{c}{\left[1-\left(\lambda_{4} / \lambda_{e}\right)^{2}\right]^{1 / 2}} $$ Note that this exceeds the velocity of light \(c !\) Find the group velocity \(c_{\theta}=d \omega / d k_{x}\) and show that $$ c_{p} c_{g}=c^{2} $$ Explain the distinction between \(c_{\rho}, c_{,}\)and \(c_{p}\) in terms of the plane-uave analysis of Prob. \(8.7 .6\) for the \(\mathrm{TE}_{10}\) mode in rectangular waveguide.

Use the results of Prob. 8.2.3 to compute the Poynting vector for a coaxial transmission line. Integrate it over the annular area between conductors and show that the power carried down the line by the wave is $$ P=i^{2} Z_{0}=\frac{v^{2}}{Z_{0}}, $$ where \(i\) and \(v\) are the instantaneous current and voltage and \(Z_{0}\) is the characteristic impedance \((8.1 .9)\), that is, just the result one would expect from elementary circuit analysis.

Consider an E-field line of force, i.e., a continuous line everywhere parallel to the local direction of \(\mathbf{E}\), deflected at the boundary between two uniform media. Show that the exit line of force lies in the plane determined by the entrance line and the normal to the boundary surface and that the angles of incidence \(\theta_{1}\) and exit \(\theta_{2}\), measured with respect to the normal, are related by the Snell's law equation $$ \frac{1}{\kappa_{\theta 1}} \tan \theta_{1}=\frac{1}{K_{\theta 2}} \tan \theta_{2} \text {. } $$ What are the corresponding equations for \(\mathbf{B}, \mathbf{D}\), and \(\mathbf{H}\) ?

(a) For waves varying sinusoidally with time as \(e^{j w t}\), show that the conductivity can be eliminated from (8.5.7) and (8.5.8) by substituting for the relative permittivity the complex quantity $$ \vec{k}_{e}=\kappa_{q}-j \frac{g}{\omega t_{0}} . $$ Then all electromagnetic properties of the medium are contained in only two constants, \(\vec{x}_{e}\) and \(\kappa_{m}\). (b) When currents flow nonuniformly in space, it is possible that a net charge density Pfres builds up at certain locations, Show that the complex permittivity formalism of part (a) not only eliminates the \(\mathbf{J}_{\text {freo }}\) term in Maxwell's equation \((8.2 .20)\) but also eliminates the piros term in \((8.2 .17)\). (c) As an alternative to the formalism of part ( \(a\) ), show that the relative permittivity can be disregarded, i.e., set equal to unity, by introducing the complex conductivity $$ \ddot{g}=g+j \cot \theta\left(\kappa_{c}-1\right) \text {. } $$ In this case, the properties of the medium are specified by the two constants \(g{g}\) and \(\kappa_{w}\) -
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