Untitled Quiz

Maxwell's Equations

• Gauss's Law for Electric Fields: The electric flux passing through any closed surface is equivalent to the total charge enclosed by that surface. This can be represented as: 𝛻.𝐷 = 𝜌𝑣.
• Gauss's Law for Magnetic Fields: Magnetic flux lines are closed and do not terminate on a "magnetic charge." This results in 𝛻.𝐵 = 0.
• Ampere's Law: The line integral of H about any closed path is exactly equal to the direct current enclosed by that path, expressed as
• ර 𝑯.𝒅𝒍 = 𝐼𝑒𝑛𝑐.
• Faraday's Law: A time-varying magnetic field produces an electromotive force (emf) that may establish a current in a suitable closed circuit.

Electromagnetic Fields

• Magnetic Flux: Can be represented mathematically as: 𝜑 = ඵ 𝑩.𝒅𝒔.

Ampere's Law with Displacement Current

• Ampere's Law (with displacement current) states:
• ර 𝐻.𝑑𝑙 = ඵ 𝐽𝑑.𝒅𝒔 + ඵ 𝐽𝑐.𝒅𝒔.
• This incorporates the displacement current (J_d), which is the rate of change of electric displacement field, and conduction current (J_c) which is the flow of electric charge.
• The displacement current can be mathematically represented as:
• 𝐽𝑑 = 𝜕𝐷/𝜕𝑡 and J_c = 𝜎𝐸.
• Combining these, we get:
• ර 𝐻.𝑑𝑙 = ඵ 𝜕𝐷/𝜕𝑡.𝒅𝒔 + ඵ 𝜎𝐸.𝒅𝒔.
• Applying Stoke's theorem, we can express this as:
• 𝛻𝑥 𝐻 = 𝜕𝐷/𝜕𝑡 + 𝜎𝐸, which can be further simplified as: 𝛻𝑥 𝐻 = ε 𝜕𝐸/𝜕𝑡 + 𝜎𝐸.

Faraday's Law

• Faraday's Law states:
• 𝑒𝑚𝑓 = − 𝑑𝜑/𝑑𝑡, where 𝜑 is the magnetic flux.
• This is also expressed as:
• ර 𝐸.𝑑𝑙 = − 𝑑𝜑/𝑑𝑡.
• Faraday's Law in integral form:
• ර 𝐸.𝑑𝑙 = − 𝑑/𝑑𝑡 (ඵ 𝐵.𝑑𝑠).
• Using Stoke's theorem, we can represent this in differential form:
• 𝛻𝑥 𝐸 = − 𝜕𝐵/𝜕𝑡, which can be simplified as: 𝛻𝑥 𝐸 = −𝜇 𝜕𝐻/𝜕𝑡.

Maxwell's Equations Summary

• Maxwell's Equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields.

• Integral Form:

• Faraday's Law: ර 𝐸.𝑑𝑙 = − 𝑑/𝑑𝑡 (ඵ 𝐵.𝑑𝑠).

• Ampere's Law: ර 𝐻.𝑑𝑙 = ඵ 𝜕𝐷/𝜕𝑡.𝒅𝒔 + ඵ 𝜎𝐸.𝒅𝒔.

• Gauss's Law for Electric Fields: 𝛻.𝐷 = 𝜌𝑣.

• Gauss's Law for Magnetic Fields: 𝛻.𝐵 = 0.

• Differential Form:

• Faraday's Law:𝛻𝑥 𝐸 = −𝜇 𝜕𝐻/𝜕𝑡.

• Ampere's Law: 𝛻𝑥 𝐻 = ε 𝜕𝐸/𝜕𝑡 + 𝜎𝐸.

• Gauss's Law for Electric Fields: 𝛻.𝐷 = 𝜌𝑣.

• Gauss's Law for Magnetic Fields: 𝛻.𝐵 = 0.

• Stock's theorem and Divergence theorems are essential tools used to derive the differential form of Maxwell's equations from their integral form.

Displacement Current

• In a medium with no conductivity (𝜎=0), Maxwell’s equation becomes (𝛻𝑥 𝐻 = 𝜕𝐷/𝜕𝑡).
• This signifies the significance of displacement current in understanding the behavior of electromagnetic fields, particularly in regions with changing electric fields, even without the presence of conduction current.