Introduction to Displacement Current

• Displacement current, denoted as $I_d$ , is a concept introduced by James Clerk Maxwell in his modification to Ampere’s circuital law.
• It is not an actual flow of charge but a term included in Ampere’s law to account for the changing electric field.
• Displacement current plays a crucial role in the electromagnetic wave propagation and the generation of magnetic fields.

Ampere’s Circuital Law

• Ampere’s circuital law relates the magnetic field around a closed loop to the electric current flowing through the loop.
• Mathematically, it can be written as: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot \mu_r \cdot I_{\text{enc}}$ where:
  • $\vec{B}$ is the magnetic field
  • $d\vec{l}$ is an infinitesimal element of the closed loop
  • $\mu_0$ is the permeability of free space
  • $\mu_r$ is the relative permeability of the medium
  • $I_{\text{enc}}$ is the total current passing through the surface enclosed by the loop

Maxwell’s Correction to Ampere’s Law

• Ampere’s circuital law was found inadequate to explain certain electromagnetic phenomena, such as the behavior of capacitors and the propagation of electromagnetic waves.
• Maxwell introduced a correction term to Ampere’s law, known as displacement current, to account for the changing electric field.
• The modified form of Ampere’s law, including the displacement current, is given by: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot \mu_r \cdot (I_{\text{enc}} + I_d)$

The Need for Displacement Current

• Displacement current arises from the time-varying electric field in a region of space.
• It is essential to include displacement current in Ampere’s law to maintain the conservation of charge and to accurately describe the behavior of electromagnetic waves.
• Displacement current enables electromagnetic waves to propagate through space and explains the interaction between electric and magnetic fields.

Relationship between Electric Current and Displacement Current

• The relationship between electric current ( $I$ ) and displacement current ( $I_d$ ) can be understood using Gauss’s law of electrostatics.
• Gauss’s law states that the divergence of the electric field ( $\nabla \cdot \vec{E}$ ) is proportional to the charge density ( $\rho$ ). $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$
• When there is a time-varying magnetic field, it induces an electric field according to Faraday’s law of electromagnetic induction.
• The changing electric field contributes to the divergence of the electric field and gives rise to the displacement current.

11. Displacement Current - Example Problem

• Consider a parallel plate capacitor with a changing electric field. The electric field between the plates is given by $E = \frac{dV}{dt}$ , where $V$ is the voltage across the plates.
• Using Ampere’s law, we can calculate the displacement current flowing through the plates. Let’s assume the surface area enclosed by the loop is $A$ and the changing electric field is uniform within this area.
• The equation for Ampere’s law with displacement current is: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot \mu_r \cdot (I_{\text{enc}} + I_d)$
• For a parallel plate capacitor, the magnetic field lines are perpendicular to the loop enclosing the plates. Hence, $\vec{B} \cdot d\vec{l} = B \cdot dl$ .
• In this case, the displacement current ( $I_d$ ) is given by: $I_d = \varepsilon_0 \cdot \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$
• By substituting the values and integrating over the surface area, we can find the displacement current flowing through the plates.

12. Applications of Displacement Current

• Displacement current has several practical applications in various fields, including:
  • Wireless communication: Displacement current plays a crucial role in the generation and transmission of electromagnetic waves used in wireless communication systems.
  • Capacitors: Displacement current explains the charging and discharging behavior of capacitors and their ability to store and release electrical energy.
  • Radiation and antennas: Displacement current is fundamental to the generation and radiation of electromagnetic waves by antennas, enabling communication through radio, television, and satellite signals.
  • Electromagnetic compatibility: Understanding displacement current helps engineers design electronic devices and systems that are immune to interference from electromagnetic fields.

13. Relationship between Displacement Current and Time-Varying Electric Field

• The displacement current ( $I_d$ ) arises due to the changing electric field ( $\vec{E}$ ) in a region.
• According to Faraday’s law of electromagnetic induction, a changing magnetic field induces an electric field ( $\vec{E}$ ):
  • $\vec{E} = -\frac{d\vec{B}}{dt}$
• The changing electric field ( $\vec{E}$ ) contributes to the divergence of the electric field ( $\nabla \cdot \vec{E}$ ), which gives rise to the displacement current ( $I_d$ ):
  • $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0} + \frac{d}{dt}(\nabla \cdot \vec{B})$
• The displacement current is an essential component to maintain the continuity equation and the conservation of charge in electromagnetic systems.

14. Calculation of Displacement Current

• To calculate the displacement current flowing through a particular surface, the following steps can be followed:
  1. Determine the surface that encloses the region of interest.
  2. Calculate the changing electric field within this surface.
  3. Use the equation for displacement current: $I_d = \varepsilon_0 \cdot \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$ .
  4. Evaluate the integral of $\vec{E} \cdot d\vec{A}$ over the surface.
  5. Differentiate the result with respect to time to obtain the time rate of change of the electric flux.
  6. Finally, multiply by $\varepsilon_0$ to find the displacement current.

15. Displacement Current vs. Conduction Current

• Displacement current ( $I_d$ ) and conduction current ( $I_c$ ) are two distinct types of current in electromagnetism.
• Conduction current refers to the flow of electric charge through a conducting medium, such as the movement of electrons in a wire. It obeys Ohm’s law and experiences resistance.
• Displacement current, on the other hand, is not related to the physical movement of electric charge but to the changing electric field. It is essential to maintain the conservation of charge and explain the behavior of electromagnetic waves.
• Displacement current does not experience resistance and is primarily associated with capacitors and the propagation of electromagnetic waves.

16. Displacement Current in a Capacitor

• Displacement current is especially significant in the behavior of capacitors.
• When a voltage is applied across the plates of a capacitor, the electric field between them changes.
• This changing electric field induces a displacement current, which complements the conduction current flowing through the capacitor’s wires.
• As the displacement current charges or discharges the plates, the electric field and voltage across the capacitor change accordingly.
• Displacement current plays a vital role in the energy storage and subsequent release in a capacitor.

17. Displacement Current and Electromagnetic Waves

• Displacement current plays a crucial role in the generation, propagation, and interaction of electromagnetic waves.
• Electromagnetic waves consist of changing electric and magnetic fields that perpetuate through space.
• The time-varying electric field induces a magnetic field, and vice versa, in a self-sustaining manner.
• Displacement current accounts for the changing electric field and enables the propagation of electromagnetic waves, even in regions where no conduction current is present.
• Without displacement current, electromagnetic waves and phenomena such as radio communication and optics would not be possible.

18. Displacement Current and Maxwell’s Equations

• Maxwell’s equations are a set of fundamental equations that describe the behavior and interaction of electric and magnetic fields.
• The inclusion of the displacement current allows Maxwell’s equations to be complete and consistent with experimental observations.
• The displacement current term appears in two of Maxwell’s equations:
  1. Ampere’s law with Maxwell’s correction: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot \mu_r \cdot (I_{\text{enc}} + I_d)$
  2. Faraday’s law of electromagnetic induction: $\oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{A}$
• These equations, along with the others, form the foundation of electromagnetism and explain a wide range of phenomena.

19. Displacement Current and Conservation of Charge

• Displacement current is essential to maintain the conservation of charge in electromagnetism.
• The continuity equation expresses the conservation of charge, stating that the divergence of the electric current density ( $\nabla \cdot \vec{J}$ ) is equal to the rate of change of charge density ( $\frac{d\rho}{dt}$ ).
• Including the displacement current in Ampere’s law is necessary to satisfy the continuity equation.
• Failure to consider the displacement current would violate the conservation of charge and lead to inconsistencies in the description of electromagnetic phenomena.

20. Conclusion

• Displacement current, introduced by Maxwell as a modification to Ampere’s law, is a crucial concept in electromagnetism.
• It accounts for the changing electric field and enables the understanding of electromagnetic wave propagation, charging behavior of capacitors, and the interaction of electric and magnetic fields.
• Displacement current is incorporated into Maxwell’s equations to ensure their consistency with experimental observations and to maintain the conservation of charge.

21. Displacement Current - Example Problem

• Consider a parallel plate capacitor with a changing electric field.
• The electric field between the plates is given by $E = \frac{dV}{dt}$ , where $V$ is the voltage across the plates.
• Using Ampere’s law, we can calculate the displacement current flowing through the plates.
• Let’s assume the surface area enclosed by the loop is $A$ and the changing electric field is uniform within this area.
• The equation for Ampere’s law with displacement current is: $\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot (I_{\text{enc}} + I_d)$

22. Displacement Current - Example Problem (continued)

• For a parallel plate capacitor, the magnetic field lines are perpendicular to the loop enclosing the plates, so $\vec{B} \cdot d\vec{l} = B \cdot dl$ .
• In this case, the displacement current ( $I_d$ ) is given by: $I_d = \varepsilon_0 \cdot \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$
• By substituting the values and integrating over the surface area, we can find the displacement current flowing through the plates.

23. Displacement Current - Example Problem (continued)

• Let’s assume the area of the loop is $A$ and the electric field between the plates is $E = \frac{dV}{dt}$ .
• The integral of $\vec{E} \cdot d\vec{A}$ over the surface area can be simplified as: $\int \vec{E} \cdot d\vec{A} = E \cdot A$
• Differentiating with respect to time, we get: $\frac{d}{dt}(E \cdot A) = \frac{dE}{dt} \cdot A = \frac{1}{\varepsilon_0} \cdot \frac{dV}{dt} \cdot A$

24. Displacement Current - Example Problem (continued)

• Substituting the result into the equation for displacement current: $I_d = \varepsilon_0 \cdot \frac{d}{dt} \int \vec{E} \cdot d\vec{A} = \varepsilon_0 \cdot \frac{1}{\varepsilon_0} \cdot \frac{dV}{dt} \cdot A = \frac{dV}{dt} \cdot A$
• The displacement current ( $I_d$ ) flowing through the plates is equal to the rate of change of voltage ( $\frac{dV}{dt}$ ) multiplied by the surface area ( $A$ ).

25. Displacement Current - Example Problem (continued)

• Let’s consider a parallel plate capacitor with a changing voltage of 10 V over a time interval of 2 seconds.
• The surface area between the plates is 0.5 m $^2$ .
• The displacement current ( $I_d$ ) can be calculated as follows: $I_d = \frac{dV}{dt} \cdot A = \frac{10 ; \text{V}}{2 ; \text{s}} \cdot 0.5 ; \text{m}^2 = 2.5 ; \text{A}$
• Therefore, the displacement current flowing through the capacitor is 2.5 Amperes.

26. Applications of Displacement Current

• Displacement current has several practical applications in various fields, including:
  • Wireless communication: Displacement current plays a crucial role in the generation and transmission of electromagnetic waves used in wireless communication systems.
  • Capacitors: Displacement current explains the charging and discharging behavior of capacitors and their ability to store and release electrical energy.
  • Radiation and antennas: Displacement current is fundamental to the generation and radiation of electromagnetic waves by antennas, enabling communication through radio, television, and satellite signals.
  • Electromagnetic compatibility: Understanding displacement current helps engineers design electronic devices and systems that are immune to interference from electromagnetic fields.
  • Magnetic levitation: Displacement current contributes to the magnetic fields involved in magnetic levitation systems, such as Maglev trains.

27. Calculation of Displacement Current

• To calculate the displacement current flowing through a particular surface, the following steps can be followed:
  1. Determine the surface that encloses the region of interest.
  2. Calculate the changing electric field within this surface.
  3. Use the equation for displacement current: $I_d = \varepsilon_0 \cdot \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$ .
  4. Evaluate the integral of $\vec{E} \cdot d\vec{A}$ over the surface.
  5. Differentiate the result with respect to time to obtain the time rate of change of the electric flux.
  6. Finally, multiply by $\varepsilon_0$ to find the displacement current.

28. Displacement Current vs. Conduction Current

• Displacement current ( $I_d$ ) and conduction current ( $I_c$ ) are two distinct types of current in electromagnetism.
• Conduction current refers to the flow of electric charge through a conducting medium, such as the movement of electrons in a wire.
• Displacement current, on the other hand, is not related to the physical movement of electric charge but to the changing electric field.
• Displacement current is essential to maintain the conservation of charge and explain the behavior of electromagnetic waves, while conduction current experiences resistance and is associated with the movement of charge carriers.

29. Displacement Current in a Capacitor

• Displacement current is especially significant in the behavior of capacitors.
• When a voltage is applied across the plates of a capacitor, the electric field between them changes.
• This changing electric field induces a displacement current, which complements the conduction current flowing through the capacitor’s wires.
• As the displacement current charges or discharges the plates, the electric field and voltage across the capacitor change accordingly.
• Displacement current plays a vital role in the energy storage and subsequent release in a capacitor.

30. Displacement Current and Electromagnetic Waves

• Displacement current plays a crucial role in the generation, propagation, and interaction of electromagnetic waves.
• Electromagnetic waves consist of changing electric and magnetic fields that perpetuate through space.
• The time-varying electric field induces a magnetic field, and vice versa, in a self-sustaining manner.
• Displacement current accounts for the changing electric field and enables the propagation of electromagnetic waves, even in regions where no conduction current is present.
• Without displacement current, electromagnetic waves and phenomena such as radio communication and optics would not be possible.