Electromagnetic Waves

8.1 INTRODUCTION

In earlier chapters, we learned that an electric current produces a magnetic field and that two current-carrying wires exert forces on each other due to their magnetic fields. Additionally, a changing magnetic field produces an electric field, as seen in Faraday's law of electromagnetic induction. The question arises: does a changing electric field also produce a magnetic field? James Clerk Maxwell answered this affirmatively, proposing that not only electric currents but also time-varying electric fields generate magnetic fields. While applying Ampere's circuital law to a charging capacitor, Maxwell noticed an inconsistency. The law, as originally formulated, did not account for the magnetic field in the region between the capacitor plates where no conduction current flows. To resolve this, Maxwell introduced the concept of displacement current, an additional current term arising from the changing electric field. This addition made Ampere's law consistent and symmetric with Faraday's law. Maxwell formulated a set of four equations, now known as Maxwell's equations, that unify electricity, magnetism, and optics. These equations describe how electric and magnetic fields are generated by charges, currents, and changes in each other. One of the most profound predictions from Maxwell's equations is the existence of electromagnetic waves—coupled oscillations of electric and magnetic fields that propagate through space at a speed very close to the speed of light (3 × 10^8 m/s). This led to the remarkable conclusion that light itself is an electromagnetic wave, unifying the previously separate domains of electricity, magnetism, and optics. Heinrich Hertz experimentally demonstrated electromagnetic waves in 1885, confirming Maxwell's prediction. Subsequently, the technological exploitation of these waves revolutionized communication. This chapter begins by discussing the need for displacement current and its implications, followed by a qualitative description of electromagnetic waves and their properties. Finally, the chapter covers the electromagnetic spectrum, spanning from gamma rays with extremely short wavelengths to long radio waves with very large wavelengths.

• Electric currents produce magnetic fields; changing magnetic fields produce electric fields.
• Maxwell introduced displacement current to resolve inconsistencies in Ampere's law.
• Maxwell's equations unify electricity, magnetism, and optics.
• Electromagnetic waves are coupled oscillations of electric and magnetic fields propagating through space.
• Speed of electromagnetic waves equals the speed of light, leading to the conclusion that light is an electromagnetic wave.
• Hertz experimentally confirmed electromagnetic waves in 1885.
• 📌 Displacement current: A current term introduced by Maxwell due to time-varying electric fields, acting as a source of magnetic field.
• 📌 Maxwell's equations: A set of four fundamental equations describing electromagnetism.
• 📌 Electromagnetic waves: Self-sustaining oscillations of electric and magnetic fields propagating through space.

8.2 DISPLACEMENT CURRENT

Ampere's circuital law states that the line integral of the magnetic field B around a closed loop equals μ₀ times the conduction current passing through any surface bounded by the loop. However, when applied to a charging capacitor, this law leads to a contradiction. Consider a parallel plate capacitor connected to a time-varying current i(t). Outside the capacitor, the conduction current flows in the wires, producing a magnetic field B at a point P. Using Ampere's law with a loop encircling the wire, the magnetic field is given by B(2πr) = μ₀ i(t), where r is the radius of the circular loop. Now, consider a different surface bounded by the same loop but passing between the capacitor plates where no conduction current flows. Ampere's law then predicts zero magnetic field, contradicting the previous result. The resolution lies in recognizing that the changing electric field between the capacitor plates acts like a current, called the displacement current. The electric field E between the plates is related to the charge Q on the plates as E = Q/(ε₀ A), where A is the plate area and ε₀ is the permittivity of free space. The electric flux through the surface between the plates is Φ_E = E × A = Q/ε₀. Differentiating with respect to time, dΦ_E/dt = (1/ε₀) dQ/dt = i/ε₀. Multiplying both sides by ε₀, we get ε₀ dΦ_E/dt = i, which identifies the displacement current i_d = ε₀ dΦ_E/dt. Thus, the total current passing through any surface bounded by the loop is the sum of the conduction current i_c and the displacement current i_d. Ampere's law is generalized to the Ampere-Maxwell law: ∮ B · dl = μ₀ i_c + μ₀ ε₀ dΦ_E/dt This generalization restores consistency and symmetry in Maxwell's equations. The displacement current produces magnetic fields just like conduction current. Inside the capacitor, conduction current is zero but displacement current is non-zero; outside the capacitor, conduction current exists but displacement current is zero. This concept has profound implications, including the prediction of electromagnetic waves. The symmetry between the laws of induction (changing magnetic fields produce electric fields) and the Ampere-Maxwell law (changing electric fields produce magnetic fields) implies that time-varying electric and magnetic fields sustain each other and propagate through space as waves.

• Ampere's circuital law without displacement current leads to contradictions in charging capacitors.
• Displacement current arises from time-varying electric flux between capacitor plates.
• Displacement current i_d = ε₀ dΦ_E/dt acts as a source of magnetic field like conduction current.
• Generalized Ampere-Maxwell law includes both conduction and displacement currents.
• Displacement current restores symmetry between electric and magnetic fields in Maxwell's equations.
• Displacement current enables the existence of electromagnetic waves.
• 📌 Conduction current (i_c): Current due to flow of charges in conductors.
• 📌 Displacement current (i_d): Current equivalent due to changing electric field, i_d = ε₀ dΦ_E/dt.
• 📌 Ampere-Maxwell law: Generalized Ampere's law including displacement current.