Introduction to Electromagnetic Waves and Displacement Current

Introduction

Throughout our study of electricity and magnetism, we have examined how electric charges create electric fields, and how electric currents create magnetic fields. We have also seen how changing magnetic fields can induce electric fields (and hence EMF and currents) through Faraday's Law of induction. These phenomena, initially studied as separate subjects, were unified by James Clerk Maxwell in the 19th century.

Maxwell's crucial insight, based on his reformulation of the laws of electricity and magnetism into a set of four elegant equations (Maxwell's equations), was that there is a symmetry between electric and magnetic fields. He predicted that a changing electric field should produce a magnetic field, just as a changing magnetic field produces an electric field. This missing piece completed the picture and led to the theoretical prediction of electromagnetic waves.

Electromagnetic waves are disturbances that propagate through space, consisting of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation. These waves carry energy and momentum.

The existence of electromagnetic waves was experimentally confirmed by Heinrich Hertz in 1888, about two decades after Maxwell's prediction. Hertz produced and detected radio waves, demonstrating that they had the properties predicted by Maxwell, including travelling at the speed of light.

The study of electromagnetic waves encompasses a vast spectrum, including radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. All these forms of radiation are fundamentally the same type of wave, differing only in their frequency and wavelength.

The generation and propagation of electromagnetic waves require the presence of accelerating electric charges. Static charges produce only electric fields. Steady currents produce both electric and magnetic fields (which are constant in time). But it is the acceleration of charges that leads to the emission of energy in the form of electromagnetic waves.

The introduction of the concept of displacement current was a key step taken by Maxwell to ensure the consistency of Ampere's Law with the continuity equation (which expresses the conservation of charge). This modification completed Ampere's Law and provided the theoretical basis for the generation of magnetic fields by changing electric fields, a crucial element in the generation and propagation of electromagnetic waves.

Displacement Current ($ I_d = \epsilon_0 \frac{d\Phi_E}{dt} $)

Ampere's Circuital Law, in its original form, states that the line integral of the magnetic field around any closed loop is proportional to the total conduction current passing through the area enclosed by the loop: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{conduction}$. While this law is valid for steady currents, Maxwell realised it was inconsistent for time-varying currents.

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Inconsistency of Ampere's Law for Time-Varying Fields

Consider a circuit where a capacitor is being charged by a current $I$. Apply the original Ampere's Law to a closed loop around the wire leading to one plate of the capacitor.

Ampere's Law applied to a charging capacitor. The same loop encloses different current depending on the surface chosen.

Let the Amperian loop be a circle C around the wire. According to Ampere's Law, $\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{conduction}$. If we choose a flat surface $S_1$ bounded by the loop C (e.g., the flat circle across the wire), the conduction current $I$ passes through this surface. So, $\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I$.

Now, consider another surface $S_2$ bounded by the same loop C, but this surface passes between the capacitor plates. As the capacitor charges, current flows into one plate and out of the other, but no conduction current flows through the gap between the plates (assuming a perfect vacuum or insulator). Therefore, the conduction current passing through surface $S_2$ is zero. According to the original Ampere's Law, $\oint_C \vec{B} \cdot d\vec{l} = \mu_0 (0) = 0$.

This leads to a contradiction: the line integral of the magnetic field around the same closed loop gives different values ($\mu_0 I$ and 0), depending on the surface chosen. This inconsistency arises when dealing with time-varying electric fields (like the field between the charging capacitor plates).

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Maxwell's Addition: Displacement Current

To resolve this inconsistency, Maxwell proposed that a changing electric field should also be a source of a magnetic field, just like a conduction current is. He introduced the concept of displacement current ($I_d$).

Consider the region between the capacitor plates. As the capacitor charges, the electric field $\vec{E}$ between the plates changes with time. The electric field between parallel plates is approximately $E = \sigma/\epsilon_0 = Q/(A\epsilon_0)$, where $\sigma$ is the surface charge density on the plates, $Q$ is the charge on the plates, and $A$ is the area of the plates.

The electric flux ($\Phi_E$) through the surface $S_2$ between the plates is $\Phi_E = \int \vec{E} \cdot d\vec{A}$. If the field is uniform and perpendicular to the area $A$, $\Phi_E = E A = \frac{Q}{A\epsilon_0} A = \frac{Q}{\epsilon_0}$.

The rate of change of electric flux is $\frac{d\Phi_E}{dt} = \frac{1}{\epsilon_0} \frac{dQ}{dt}$.

The conduction current $I$ charging the capacitor is $I = dQ/dt$. So, $\frac{d\Phi_E}{dt} = \frac{I}{\epsilon_0}$, which means $I = \epsilon_0 \frac{d\Phi_E}{dt}$.

Maxwell hypothesised that this term $\epsilon_0 \frac{d\Phi_E}{dt}$ acts as a 'current' even in vacuum or insulator and is a source of magnetic field. He called it the displacement current ($I_d$):

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

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Maxwell-Ampere Law (Generalized Ampere's Law)

Maxwell modified Ampere's original law by adding the displacement current term to the conduction current. The modified law, known as the Maxwell-Ampere Law or Generalized Ampere's Law, states that the line integral of the magnetic field around any closed loop is proportional to the sum of the conduction current and the displacement current passing through the area enclosed by the loop:

$ \oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{conduction} + I_{displacement}) $

Substituting the expression for displacement current:

$ \oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{conduction} + \epsilon_0 \frac{d\Phi_E}{dt}) $

Where $I_{conduction}$ is the net conduction current passing through the surface bounded by the loop, and $\Phi_E = \int_S \vec{E} \cdot d\vec{A}$ is the electric flux through the same surface.

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Consistency with Continuity Equation

This modified law resolves the inconsistency with the charging capacitor. For surface $S_1$ (across the wire), $I_{conduction} = I$ and the electric field is zero (or constant, so $d\Phi_E/dt = 0$). $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$. For surface $S_2$ (between the plates), $I_{conduction} = 0$, but there is a changing electric field, leading to displacement current $I_d = \epsilon_0 \frac{d\Phi_E}{dt} = \epsilon_0 \frac{d(Q/\epsilon_0)}{dt} = \frac{dQ}{dt} = I$. So $\oint \vec{B} \cdot d\vec{l} = \mu_0 (0 + I) = \mu_0 I$. The result is now consistent for both surfaces.

The displacement current is not a current of moving charges; it is associated with a changing electric field. However, it acts as a source of magnetic field just like a conduction current does. The introduction of displacement current implies that electric and magnetic fields are interdependent and can induce each other even in vacuum, paving the way for the concept of electromagnetic waves.

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Answer:

Given:

Radius of plates, $r = 5 \, cm = 0.05 \, m$

Area of plates (assuming circular), $A = \pi r^2 = \pi (0.05 \, m)^2 = \pi (0.0025) \, m^2 = 2.5\pi \times 10^{-3} \, m^2$

Rate of increase of electric field, $\frac{dE}{dt} = 10^6 \, V/(m \cdot s)$

Permittivity of free space, $\epsilon_0 = 8.854 \times 10^{-12} \, C^2/(N \cdot m^2)$

The displacement current ($I_d$) is given by $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$.

The electric flux ($\Phi_E$) between the plates is $\Phi_E = E A$ (assuming uniform field perpendicular to the plates).

So, $\frac{d\Phi_E}{dt} = \frac{d(EA)}{dt} = A \frac{dE}{dt}$ (since area $A$ is constant).

Substitute this into the formula for displacement current:

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

Substitute the given values:

$ I_d = (8.854 \times 10^{-12} \, C^2/(N \cdot m^2)) \times (2.5\pi \times 10^{-3} \, m^2) \times (10^6 \, V/(m \cdot s)) $

$ I_d = (8.854 \times 2.5\pi) \times 10^{-12} \times 10^{-3} \times 10^6 \, (C^2/(N \cdot m^2) \cdot m^2 \cdot V/(m \cdot s)) $

$ I_d = (22.135\pi) \times 10^{-9} \, (C^2 \cdot V/(N \cdot m \cdot s)) $ (Unit check: $N \cdot m = J$, $V = J/C$, so $C^2 \cdot (J/C) / (J/s) = C \cdot J \cdot s / J = C \cdot s = A$)

$ I_d \approx (22.135 \times 3.14159) \times 10^{-9} \, A $

$ I_d \approx 69.54 \times 10^{-9} \, A = 69.54 \, nA $

The displacement current between the plates is approximately 69.54 nanoamperes. This displacement current produces a magnetic field in the region between the plates, just like a conduction current would.