Chapter Notes

Introduction

In previous studies, we learned two fundamental principles of electromagnetism:

1. An electric current creates a magnetic field.
2. A magnetic field that changes with time creates an electric field.

This raises a symmetrical question: Does an electric field that changes with time also create a magnetic field? James Clerk Maxwell proposed that it does. He identified an inconsistency in Ampere's circuital law when applied to a charging capacitor and introduced a new concept to fix it: the displacement current.

Maxwell's work culminated in a set of four fundamental equations, known as Maxwell's equations, which, along with the Lorentz force formula, form the complete foundation of electromagnetism.

One of the most profound predictions of these equations was the existence of electromagnetic waves. These are coupled, time-varying electric and magnetic fields that travel through space. Maxwell calculated the speed of these waves and found it to be approximately $3 \times 10^{8} \text{ m/s}$, which was remarkably close to the measured speed of light. This led to the revolutionary conclusion that light itself is an electromagnetic wave, unifying the fields of electricity, magnetism, and optics.

This discovery was experimentally confirmed by Hertz in 1885 and later harnessed for communication by Marconi, setting the stage for modern technology.

Displacement Current

Maxwell discovered that for logical consistency, a changing electric field must produce a magnetic field. This is a crucial concept that explains the existence of all forms of electromagnetic waves, from radio waves to gamma rays.

The Inconsistency in Ampere's Law

Let's consider a parallel plate capacitor being charged by a time-varying current, $i(t)$. Ampere's circuital law is given by: $\oint \mathbf{B} \cdot \mathbf{d} \mathbf{l}=\mu_{0} i(t)$ To find the magnetic field at a point P outside the capacitor, we can draw a circular loop around the wire. The law works perfectly here, giving a non-zero magnetic field.

However, the law states that for a given closed loop, any surface that has that loop as its boundary should give the same result.

• Surface 1 (Flat Disc): A flat surface pierced by the current-carrying wire gives a magnetic field, since current i passes through it.
• Surface 2 (Pot-like Surface): Imagine a surface shaped like a pot, where the rim is our loop, but the bottom of the pot is between the capacitor plates (see Figure 8.1b/c in the source). No conduction current (flow of charges) passes through this surface.

This leads to a contradiction:

• Using Surface 1, Ampere's law gives a magnetic field at P.
• Using Surface 2, Ampere's law gives zero magnetic field at P.

Since Ampere's law gives two different answers for the same point, the law must be incomplete.

Maxwell's Solution

Maxwell realized that although no charges flow between the capacitor plates, something else is present and changing: the electric field.

1. The electric field E between the plates is related to the charge Q on the plates by $E = Q / (\varepsilon_{0}A)$.
2. The electric flux $\Phi_{E}$ through a surface of area A between the plates is $\Phi_{\mathrm{E}}=|\mathbf{E}| A = \frac{Q}{\varepsilon_{0}}$.
3. As the capacitor charges, Q changes with time, which means the electric flux $\Phi_{E}$ also changes. The rate of change is: $\frac{\mathrm{d} \Phi_{E}}{\mathrm{~d} t} = \frac{1}{\varepsilon_{0}} \frac{\mathrm{d} Q}{\mathrm{~d} t}$
4. Since the current flowing into the capacitor is $i = \frac{\mathrm{d} Q}{\mathrm{~d} t}$, we can write: $i = \varepsilon_{0} \frac{\mathrm{d} \Phi_{E}}{\mathrm{~d} t}$

This term, $\varepsilon_{0} \frac{\mathrm{d} \Phi_{E}}{\mathrm{~d} t}$, is the missing piece. Maxwell called it the displacement current, denoted by $i_d$. It's not a current made of moving charges but is a current produced by a time-varying electric field.

The total current, $i$, is the sum of the conduction current ($i_c$, due to flowing charges) and the displacement current ($i_d$). $i=i_{c}+i_{d}=i_{c}+\varepsilon_{0} \frac{\mathrm{d} \Phi_{E}}{\mathrm{~d} t}$

• In the wires leading to the capacitor, the current is entirely conduction current ($i = i_c$).
• In the gap between the capacitor plates, the current is entirely displacement current ($i = i_d$). The value of the current is the same in both regions, maintaining continuity.

Ampere-Maxwell Law

Maxwell modified Ampere's law to include the displacement current. The generalized and correct form is known as the Ampere-Maxwell law: $\oint \mathbf{B} \cdot \mathrm{d} \mathbf{l}=\mu_{0} i_{c}+\mu_{0} \varepsilon_{0} \frac{\mathrm{~d} \Phi_{E}}{\mathrm{~d} t}$ This law states that a magnetic field can be produced by both a conduction current and a time-varying electric flux.

This discovery created a beautiful symmetry in electromagnetism:

• A changing magnetic field creates an electric field (Faraday's Law).
• A changing electric field creates a magnetic field (Ampere-Maxwell Law).

This mutual creation and regeneration of electric and magnetic fields is the very essence of an electromagnetic wave.

Maxwell's Equations in Vacuum

These are the four fundamental equations that govern all electric and magnetic phenomena.

1. Gauss's Law for electricity: $\oint \mathbf{E} \cdot \mathrm{d} \mathbf{A}=Q / \varepsilon_{0}$ (Relates electric field to the charge that creates it)
2. Gauss's Law for magnetism: $\oint \mathbf{B} \cdot \mathrm{d} \mathbf{A}=0$ (States that there are no magnetic monopoles)
3. Faraday's Law: $\oint \mathbf{E} \cdot \mathrm{d} \mathbf{l}=\frac{-\mathrm{d} \Phi_{\mathrm{B}}}{\mathrm{d} t}$ (A changing magnetic flux creates an electric field)
4. Ampere-Maxwell Law: $\oint \mathbf{B} \cdot \mathrm{d} \mathbf{l}=\mu_{0} \mathrm{i}_{\mathrm{c}}+\mu_{0} \varepsilon_{0} \frac{\mathrm{d} \Phi_{\mathrm{E}}}{\mathrm{d} t}$ (A magnetic field is created by a conduction current and/or a changing electric flux)

Electromagnetic Waves

Sources of Electromagnetic Waves

How are these waves produced?

• Stationary charges produce only static electric fields.
• Charges in uniform motion (a steady current) produce static magnetic fields.
• Accelerated charges radiate electromagnetic waves.

An oscillating charge is a perfect example of an accelerating charge. It produces an oscillating electric field, which in turn produces an oscillating magnetic field, which regenerates the electric field, and so on. These mutually sustaining fields propagate through space as an electromagnetic wave. The frequency of the wave is the same as the frequency of the oscillating charge.

Nature of Electromagnetic Waves

From Maxwell's equations, we can deduce the key properties of electromagnetic waves:

• They are transverse waves, meaning the electric field ($\mathbf{E}$) and magnetic field ($\mathbf{B}$) are perpendicular to each other and also perpendicular to the direction of wave propagation.
• For a wave traveling in the z-direction, the fields can be described by sinusoidal functions: $E_{x}=E_{0} \sin (k z-\omega t)$ $B_{y}=B_{0} \sin (k z-\omega t)$ Where $E_0$ and $B_0$ are the amplitudes, $k$ is the wave number ($k = 2\pi/\lambda$), and $\omega$ is the angular frequency ($\omega = 2\pi\nu$).
• The amplitudes of the electric and magnetic fields are related by: $B_{0} = \frac{E_{0}}{c}$
• The speed of electromagnetic waves in a vacuum (free space) is a fundamental constant, c, determined by the permittivity ($\varepsilon_0$) and permeability ($\mu_0$) of free space: $c = \frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}} \approx 3 \times 10^8 \text{ m/s}$
• The relationship between frequency ($\nu$), wavelength ($\lambda$), and speed (c) is: $\nu \lambda = c$
• Unlike mechanical waves (like sound), electromagnetic waves do not require a material medium to travel. They can propagate through the vacuum of space.
• When an electromagnetic wave travels through a material medium with permittivity $\varepsilon$ and permeability $\mu$, its speed v is reduced: $v = \frac{1}{\sqrt{\mu \varepsilon}}$
• Electromagnetic waves carry energy from one place to another. This is how the sun's energy reaches Earth and how radio signals are transmitted.

Given

• Electric field, $\mathbf{E}=6.3 \hat{\mathbf{j}} \mathrm{~V} / \mathrm{m}$
• Speed of light in free space, $c = 3 \times 10^8 \text{ m/s}$
• Direction of propagation is along the x-axis ($\hat{\mathbf{i}}$)
• Direction of $\mathbf{E}$ is along the y-axis ($\hat{\mathbf{j}}$)

To Find

The magnetic field vector $\mathbf{B}$

Formula

$B = \frac{E}{c}$

Solution

First, calculate the magnitude of the magnetic field: $B = \frac{6.3 \mathrm{~V} / \mathrm{m}}{3 \times 10^{8} \mathrm{~m} / \mathrm{s}}=2.1 \times 10^{-8} \mathrm{~T}$ Next, find the direction. The direction of propagation is given by the direction of $\mathbf{E} \times \mathbf{B}$. We know the propagation is along the x-axis ($\hat{\mathbf{i}}$) and $\mathbf{E}$ is along the y-axis ($\hat{\mathbf{j}}$). We need to find the direction of $\mathbf{B}$ such that $(\hat{\mathbf{j}}) \times (\text{direction of } \mathbf{B}) = \hat{\mathbf{i}}$.

From vector algebra, we know that $(+\hat{\mathbf{j}}) \times (+\hat{\mathbf{k}})=\hat{\mathbf{i}}$. Therefore, the magnetic field $\mathbf{B}$ must be along the z-direction ($\hat{\mathbf{k}}$).

Final Answer $\mathbf{B}=2.1 \times 10^{-8} \hat{\mathbf{k}}$ T

Given

• Magnetic field equation: $\mathrm{B}_{y}=\left(2 \times 10^{-7}\right) \mathrm{T} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right)$
• Standard wave equation form: $B_y = B_0 \sin(kx + \omega t)$

To Find

(a) Wavelength ($\lambda$) and frequency ($\nu$) (b) Expression for the electric field ($\mathbf{E}$)

Formula

$k = \frac{2\pi}{\lambda}$ $\omega = 2\pi\nu$ $E_0 = c B_0$

Solution

By comparing the given equation with the standard form, we can identify the components:

• Amplitude, $B_0 = 2 \times 10^{-7} \text{ T}$
• Wave number, $k = 0.5 \times 10^3 \text{ m}^{-1}$
• Angular frequency, $\omega = 1.5 \times 10^{11} \text{ rad/s}$

(a) Wavelength and Frequency

From the wave number $k$, we find the wavelength $\lambda$: $\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.5 \times 10^3} \text{ m} \approx 1.26 \times 10^{-2} \text{ m} = 1.26 \text{ cm}$

From the angular frequency $\omega$, we find the frequency $\nu$: $\nu = \frac{\omega}{2\pi} = \frac{1.5 \times 10^{11}}{2\pi} \text{ Hz} \approx 23.9 \times 10^9 \text{ Hz} = 23.9 \text{ GHz}$

Answer for part (a) = Wavelength $\lambda = 1.26$ cm, Frequency $\nu = 23.9$ GHz.

(b) Expression for the Electric Field

First, calculate the amplitude of the electric field, $E_0$: $E_0 = c B_0 = (3 \times 10^8 \text{ m/s}) \times (2 \times 10^{-7} \text{ T}) = 60 \text{ V/m}$ The electric field must be perpendicular to both the direction of propagation (x-axis) and the magnetic field (y-axis). Therefore, the electric field must be along the z-axis. The phase $(kx + \omega t)$ remains the same for both fields.

The expression for the electric field is: $E_{z}=60 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \mathrm{V} / \mathrm{m}$

Answer for part (b) = $E_{z}=60 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \mathrm{V} / \mathrm{m}$

Electromagnetic Spectrum

Electromagnetic waves exist over a vast range of frequencies and wavelengths. This continuous range is called the electromagnetic spectrum. The spectrum is broadly divided into regions based on how the waves are produced and detected, though there are no sharp boundaries between them. Here they are, in order of increasing frequency (decreasing wavelength).

Radio waves

• Wavelength: $> 0.1 \text{ m}$
• Production: Produced by the accelerated motion of charges in conducting wires (antennas).
• Uses: Radio and television communication systems, AM/FM radio, cellular phones.

Microwaves

• Wavelength: $0.1 \text{ m}$ to $1 \text{ mm}$
• Production: Special vacuum tubes like klystrons and magnetrons.
• Uses: Radar systems for aircraft navigation, speed guns used by police, and microwave ovens. In a microwave oven, the frequency is tuned to the resonant frequency of water molecules, efficiently transferring energy to heat food.

Infrared waves

• Wavelength: $1 \text{ mm}$ to $700 \text{ nm}$
• Production: Vibration of hot bodies and molecules.
• Uses: Often called "heat waves." Used in physical therapy, maintaining Earth's warmth via the greenhouse effect, infrared detectors on satellites, and remote controls for electronic devices.

Visible rays (Light)

• Wavelength: $700 \text{ nm}$ (red) to $400 \text{ nm}$ (violet)
• Production: Emitted by electrons in atoms when they move from a higher energy level to a lower one.
• Detection: The human eye, photocells, photographic film. This is the narrow band of the spectrum that we can see.

Ultraviolet (UV) rays

• Wavelength: $400 \text{ nm}$ to $1 \text{ nm}$
• Production: Special lamps and very hot bodies, like the Sun.
• Uses: Used in water purifiers to kill germs and for high-precision applications like LASIK eye surgery.
• Effects: Causes skin tanning and can be harmful in large doses. Most of the Sun's UV radiation is absorbed by the ozone layer in the atmosphere.

X-rays

• Wavelength: $1 \text{ nm}$ to $10^{-3} \text{ nm}$
• Production: Bombarding a metal target with high-energy electrons.
• Uses: Medical imaging (X-ray scans) and treatment for certain types of cancer.
• Effects: Can damage or destroy living tissue, so exposure should be minimized.

Gamma rays

• Wavelength: $< 10^{-3} \text{ nm}$
• Production: Produced in nuclear reactions and by the decay of radioactive nuclei.
• Uses: Used in medicine to destroy cancer cells (radiotherapy). They are the most energetic waves in the spectrum.