13.1 Displacement current and Ampère-Maxwell law

Maxwell's correction to Ampère's law introduced displacement current, fixing a flaw in the original equation. This addition accounts for time-varying electric fields, ensuring consistency with the continuity equation and conservation of total current.

The Ampère-Maxwell law, a fundamental equation in electromagnetism, describes the relationship between magnetic fields, electric currents, and changing electric fields. It's crucial for understanding electromagnetic waves and AC circuits.

Displacement Current and Ampère-Maxwell Law

Maxwell's Correction to Ampère's Law

• Maxwell introduced the concept of displacement current to fix a flaw in Ampère's law
• Ampère's law failed to account for time-varying electric fields, which led to inconsistencies with the continuity equation
• Maxwell's correction adds the displacement current term ($\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$) to Ampère's law
  • This term represents the rate of change of the electric field over time
  • It ensures that the total current (conduction + displacement) is conserved

Displacement Current and its Properties

• Displacement current ($I_D$) is a term added to the conduction current to obtain the total current
  • It is defined as $I_D = \epsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux
• Displacement current is not an actual flow of charges, but a time-varying electric field that induces a magnetic field
  • It occurs in capacitors, where the electric field between the plates changes with time (charging or discharging)
  • It also exists in empty space, such as in electromagnetic waves
• The displacement current density ($\mathbf{J}_D$) is related to the time-varying electric field by $\mathbf{J}_D = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Ampère-Maxwell Law and its Applications

• The Ampère-Maxwell law is the modified version of Ampère's law that includes the displacement current term
  • In integral form: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$
  • In differential form: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
• The Ampère-Maxwell law is one of the four fundamental equations of electromagnetism (Maxwell's equations)
  • It describes the relationship between magnetic fields, electric currents, and time-varying electric fields
• Applications of the Ampère-Maxwell law include:
  • Analyzing the behavior of capacitors and inductors in AC circuits
  • Explaining the propagation of electromagnetic waves in vacuum and media
  • Designing and optimizing electromagnetic devices (transformers, motors, generators)

Conduction Current and Continuity Equation

Conduction Current and Charge Conservation

• Conduction current ($I_C$) is the flow of electric charges (electrons) through a conductor
  • It is defined as the rate of flow of charge ($I_C = \frac{dQ}{dt}$)
• The conduction current density ($\mathbf{J}_C$) is related to the electric field by Ohm's law: $\mathbf{J}_C = \sigma \mathbf{E}$, where $\sigma$ is the conductivity
• The continuity equation expresses the conservation of electric charge
  • In integral form: $\frac{d}{dt} \int_V \rho dV = -\oint_S \mathbf{J} \cdot d\mathbf{A}$
  • In differential form: $\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0$

Capacitor Charging and Displacement Current

• When a capacitor is charging, the conduction current flows through the connecting wires
  • However, there is no conduction current between the capacitor plates (assuming a perfect insulator)
• The displacement current accounts for the charging of the capacitor
  • As the electric field between the plates changes, it induces a magnetic field (as per the Ampère-Maxwell law)
  • The displacement current ensures that the total current (conduction + displacement) is continuous and conserved
• The displacement current in a capacitor is equal to the conduction current in the wires connected to it
  • This maintains the continuity of current and ensures charge conservation

Relationship between Conduction Current and Displacement Current

• In a closed circuit, the conduction current and displacement current are complementary
  • The conduction current flows through conductors (wires, resistors)
  • The displacement current "flows" through insulators (capacitors, empty space)
• The total current, which is the sum of conduction and displacement currents, is always continuous and conserved
  • This is a consequence of the continuity equation and the Ampère-Maxwell law
• In electromagnetic waves, the conduction current is zero (in vacuum), and the displacement current is responsible for the propagation of the wave
  • The time-varying electric and magnetic fields sustain each other, creating a self-propagating wave