Displacement Current and Electromagnetic Waves

Displacement Current

In the study of electromagnetism, Ampère's law relates the magnetic field around a closed loop to the electric current passing through the loop. However, this law, in its original form, faces inconsistencies when applied to situations involving changing electric fields, such as in a charging capacitor. It was James Clerk Maxwell who recognized this limitation and introduced the concept of displacement current to resolve these issues, thereby unifying electricity, magnetism, and light.

Consider a parallel plate capacitor being charged by a DC source. As the capacitor charges, the electric field between the plates increases. If we apply Ampère's circuital law to a loop between the plates, we find that there is no conduction current (current due to the flow of charge carriers) passing through the surface bounded by the loop. This would imply that there is no magnetic field generated, which contradicts experimental observations.

Maxwell proposed that a changing electric flux through a surface also produces a magnetic field, just as a conduction current does. He defined this phenomenon as displacement current. The displacement current ($I_D$) is proportional to the rate of change of electric flux ($\Phi_E$) through the surface:

$I_D = \epsilon_0 \frac{d\Phi_E}{dt}$

where $\epsilon_0$ is the permittivity of free space.

The electric flux ($\Phi_E$) through a surface is given by the integral of the electric field ($\vec{E}$) over the surface area ($A$):

$\Phi_E = \int \vec{E} \cdot d\vec{A}$

For a parallel plate capacitor with a uniform electric field $\vec{E}$ and area $A$, the electric flux is $\Phi_E = EA$. If the electric field is changing with time, then the displacement current is:

$I_D = \epsilon_0 A \frac{dE}{dt}$

The total current that produces a magnetic field, according to Maxwell's modified Ampère's law, is the sum of the conduction current ($I_C$) and the displacement current ($I_D$).

Modified Ampère's Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_C + I_D)$

In the case of the charging capacitor, between the plates, there is no conduction current ($I_C = 0$), but there is a changing electric field, hence a displacement current ($I_D$). This displacement current generates a magnetic field around the loop, consistent with observations.

Significance of Displacement Current

The concept of displacement current is crucial because it establishes a symmetry between electric and magnetic fields. Just as a changing magnetic field induces an electric field (Faraday's Law of Induction), a changing electric field induces a magnetic field (Maxwell's contribution). This symmetry is fundamental to the existence and propagation of electromagnetic waves.

Electromagnetic Waves

Electromagnetic waves are self-propagating disturbances in the electromagnetic field that travel through space at the speed of light. They are produced by the acceleration of electric charges. Maxwell's equations, unified by the concept of displacement current, predicted the existence of these waves.

Maxwell's equations in free space (with no charges or currents) are:

• Gauss's Law for Electricity: $\nabla \cdot \vec{E} = 0$
• Gauss's Law for Magnetism: $\nabla \cdot \vec{B} = 0$
• Faraday's Law of Induction: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
• Ampère-Maxwell Law: $\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$

These equations show that a changing magnetic field ($\frac{\partial \vec{B}}{\partial t}$) produces a curl in the electric field ($\nabla \times \vec{E}$), and a changing electric field ($\frac{\partial \vec{E}}{\partial t}$) produces a curl in the magnetic field ($\nabla \times \vec{B}$). This continuous interplay between changing electric and magnetic fields allows electromagnetic waves to propagate through space, even in a vacuum.

In a vacuum, electromagnetic waves consist of oscillating electric and magnetic fields that are perpendicular to each other and also perpendicular to the direction of propagation. Such waves are called transverse waves.

Properties of Electromagnetic Waves

1. Transverse Nature: The electric field vector ($\vec{E}$) and the magnetic field vector ($\vec{B}$) are always perpendicular to each other and to the direction of wave propagation. If the wave travels along the x-axis, then $\vec{E}$ and $\vec{B}$ lie in the yz-plane.

2. Speed of Propagation: In free space, all electromagnetic waves travel at a constant speed, known as the speed of light ($c$). This speed is determined by the fundamental constants of electromagnetism:

$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

Numerically, $c \approx 3 \times 10^8$ meters per second.

3. Energy Transfer: Electromagnetic waves carry energy. The energy is shared equally between the electric and magnetic fields. The energy density of the electromagnetic wave is proportional to the square of the amplitude of the electric and magnetic fields.

4. Momentum Transfer: Electromagnetic waves also carry momentum. When an electromagnetic wave strikes a surface, it can exert a pressure, known as radiation pressure.

5. No Medium Required: Unlike mechanical waves (like sound waves), electromagnetic waves do not require a medium to propagate. They can travel through a vacuum.

6. Spectrum: Electromagnetic waves exist over a wide range of frequencies and wavelengths, known as the electromagnetic spectrum. This spectrum includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. All these forms of radiation are fundamentally the same phenomenon, differing only in their frequency, wavelength, and energy.

Equations for Electromagnetic Waves

Consider an electromagnetic wave propagating along the positive x-axis in free space. The electric and magnetic fields can be represented by sinusoidal functions. For a plane electromagnetic wave, the electric field and magnetic field components are perpendicular to each other and to the direction of propagation.

Let the electric field oscillate along the y-direction and the magnetic field oscillate along the z-direction.

Electric field: $E_y(x, t) = E_0 \sin(kx - \omega t)$

Magnetic field: $B_z(x, t) = B_0 \sin(kx - \omega t)$

Here:

• $E_0$ is the amplitude of the electric field.
• $B_0$ is the amplitude of the magnetic field.
• $k$ is the wave number ($k = 2\pi/\lambda$, where $\lambda$ is the wavelength).
• $\omega$ is the angular frequency ($\omega = 2\pi f$, where $f$ is the frequency).
• The term $(kx - \omega t)$ represents the phase of the wave.

The relationship between the amplitudes of the electric and magnetic fields is given by:

$E_0 = c B_0$ or $E = cB$ for instantaneous values.

The speed of the wave ($c$) is related to $\omega$ and $k$ by:

$c = \frac{\omega}{k}$

The frequency ($f$) and wavelength ($\lambda$) are related to the speed of light by:

$c = f \lambda$

Energy Density

The energy carried by an electromagnetic wave is distributed between the electric and magnetic fields. The energy density ($u$) is the energy per unit volume.

Energy density due to the electric field ($u_E$):

$u_E = \frac{1}{2} \epsilon_0 E^2$

Energy density due to the magnetic field ($u_B$):

$u_B = \frac{1}{2\mu_0} B^2$

In an electromagnetic wave, the electric and magnetic fields are oscillating, and at any point in space and time, the energy densities are equal: $u_E = u_B$.

The total energy density ($u$) of the electromagnetic wave is the sum of the energy densities of the electric and magnetic fields:

$u = u_E + u_B = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$

Since $E = cB$ and $c^2 = 1/(\mu_0 \epsilon_0)$, we have $1/\mu_0 = \epsilon_0 c^2$. Substituting $B^2 = E^2/c^2$:

$u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} \left(\frac{E^2}{c^2}\right) = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} (E^2 \mu_0 \epsilon_0) = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \epsilon_0 E^2 = \epsilon_0 E^2$

Similarly, substituting $E^2 = c^2 B^2 = B^2 / (\mu_0 \epsilon_0)$:

$u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 = \frac{1}{2} \epsilon_0 \left(\frac{B^2}{\mu_0 \epsilon_0}\right) + \frac{1}{2\mu_0} B^2 = \frac{1}{2\mu_0} B^2 + \frac{1}{2\mu_0} B^2 = \frac{1}{\mu_0} B^2$

So, the total energy density can be expressed as:

$u = \epsilon_0 E^2 = \frac{1}{\mu_0} B^2$

For a sinusoidal wave, the average energy density ($\langle u \rangle$) is half of the maximum energy density:

$\langle u \rangle = \frac{1}{2} \epsilon_0 E_0^2 = \frac{1}{2\mu_0} B_0^2$

Intensity and Poynting Vector

The intensity ($I$) of an electromagnetic wave is the average rate at which energy is transmitted per unit area. It is related to the average energy density and the speed of the wave.

$I = \langle u \rangle c$

Substituting the expression for $\langle u \rangle$:

$I = (\frac{1}{2} \epsilon_0 E_0^2) c = \frac{1}{2} \epsilon_0 c E_0^2$

Using $c = 1/\sqrt{\mu_0 \epsilon_0}$, we get $c^2 = 1/(\mu_0 \epsilon_0)$, so $\epsilon_0 = 1/(c^2 \mu_0)$.

$I = \frac{1}{2} \epsilon_0 c E_0^2 = \frac{1}{2} \left(\frac{1}{c^2 \mu_0}\right) c E_0^2 = \frac{1}{2c\mu_0} E_0^2$

Also, since $E_0 = cB_0$, $E_0^2 = c^2 B_0^2$.

$I = \frac{1}{2} \epsilon_0 c (c^2 B_0^2) = \frac{1}{2} \epsilon_0 c^3 B_0^2$

And using $E_0 = cB_0$, $I = \frac{1}{2} \epsilon_0 c (cB_0)^2 = \frac{1}{2} \epsilon_0 c^3 B_0^2$.

Using $E_0 = cB_0$ and $c = 1/\sqrt{\mu_0 \epsilon_0}$, we can also write $I$ in terms of $B_0$:

$I = \frac{1}{2\mu_0 c} B_0^2$

The Poynting vector ($\vec{S}$) is a vector quantity that describes the magnitude and direction of the flow of electromagnetic energy. It is defined as:

$\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$

The magnitude of the Poynting vector is the intensity of the wave: $S = |\vec{S}| = I$. The direction of $\vec{S}$ is the direction of wave propagation.

For a plane wave, the average magnitude of the Poynting vector is:

$\langle S \rangle = \frac{1}{2\mu_0} E_0 B_0 = \frac{1}{2\mu_0 c} E_0^2 = \frac{1}{2} \epsilon_0 c E_0^2$

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all possible frequencies (or wavelengths) of electromagnetic radiation. It is a continuous spectrum, but it is conventionally divided into several regions based on frequency or wavelength, and the properties and applications of the radiation.

Region	Approximate Frequency Range (Hz)	Approximate Wavelength Range (m)	Approximate Photon Energy (eV)	Sources and Applications
Radio Waves	< 3 x 109	> 0.1	< 1.2 x 10-5	Radio & TV broadcasting, radar, mobile phones, Wi-Fi. Generated by oscillating electric circuits.
Microwaves	3 x 109 - 3 x 1011	0.1 - 10-3	1.2 x 10-5 - 1.2 x 10-3	Microwave ovens, radar, telecommunications, satellite communication. Generated by klystrons, magnetrons.
Infrared (IR)	3 x 1011 - 4 x 1014	10-3 - 7.5 x 10-7	1.2 x 10-3 - 1.65	Remote controls, thermal imaging, night vision, optical fibers, heat radiation. Emitted by warm objects.
Visible Light	4 x 1014 - 7.5 x 1014	7.5 x 10-7 - 4 x 10-7	1.65 - 3.1	Human vision, photography, lighting, lasers. Emitted by stars, lamps, LEDs. Includes colors: Violet (400 nm), Indigo, Blue, Green, Yellow, Orange, Red (700 nm).
Ultraviolet (UV)	7.5 x 1014 - 3 x 1016	4 x 10-7 - 10-8	3.1 - 124	Sterilization, tanning beds, detecting counterfeit currency, vitamin D production. Emitted by the sun, UV lamps. Can cause skin damage.
X-rays	3 x 1016 - 3 x 1019	10-8 - 10-11	124 - 1.24 x 105	Medical imaging (radiography), security scanners, cancer therapy. Produced by decelerating high-energy electrons or electron transitions in atoms.
Gamma Rays	> 3 x 1019	< 10-11	> 1.24 x 105	Medical treatment (radiotherapy), sterilization, nuclear physics research. Emitted by radioactive decay and nuclear reactions. Highly penetrating and energetic.

Maxwell's Contribution

James Clerk Maxwell (1831-1879) was a Scottish theoretical physicist. His most significant contribution was the unification of electricity and magnetism into a single theory, described by a set of equations known as Maxwell's equations. These equations predicted the existence of electromagnetic waves and showed that light itself is an electromagnetic wave. His work laid the foundation for much of modern physics and technology, including radio, television, and telecommunications.