Electric Field - Circular Parallel Plate Capacitors

• Circular parallel plate capacitors have two circular plates with opposite charges placed parallel to each other.
• The electric field between the plates is radial and its magnitude is given by , where is the surface charge density and is the permittivity of free space.
• The electric field is directed from positive charges to negative charges.
• The magnitude of electric field decreases as we move away from the center of the circular plates.
• The electric field is zero at the center of the circular plates.

Magnetic Field - Circular Parallel Plate Capacitors

• Due to the presence of displacement current, a magnetic field exists between the circular parallel plate capacitors.
• The magnetic field follows the right-hand rule and is concentric circles around the center of the plates.
• The magnitude of the magnetic field inside the circular plates is given by , where is the permeability of free space, is the relative permeability of the medium between the plates, is the displacement current, and is the distance from the center of the plates.
• The magnetic field is stronger closer to the plates and weaker farther away.
• The direction of the magnetic field depends on the direction of the electric field and the orientation of the circular plates.

Ampere’s Circuital Law

• Ampere’s circuital law relates the magnetic field around a closed loop to the current passing through the loop.
• It states that the line integral of magnetic field around a closed loop is equal to the product of current passing through the loop and the permittivity of free space.
• Mathematically, Ampere’s law can be represented as: , where is the magnetic field, is the differential length along the closed loop, is the permeability of free space, and is the current passing through the loop.
• Ampere’s law applies to cases where the magnetic field is steady and there are no time-varying electric fields involved.

Derivation of Displacement Current

• To understand the derivation of displacement current, let’s consider a circular parallel plate capacitor with a changing electric field.
• The electric field between the plates is given by .
• The electric flux is given by , where is the area of the circular plates.
• Differentiating the electric flux with respect to time, we get .
• Substituting the value of electric field and differentiating, we obtain .
• Simplifying the equation further, we get .
• This equation shows the existence of displacement current between the circular parallel plate capacitors.

Role of Displacement Current in Maxwell’s Equations

• Displacement current plays a crucial role in Maxwell’s equations, which describe the behavior of electric and magnetic fields.
• In the absence of displacement current, only Ampere’s law could be used to describe the magnetic fields generated by steady currents.
• However, with the inclusion of displacement current, Maxwell’s equations are able to explain various electromagnetic phenomena, including the propagation of electromagnetic waves.
• Maxwell’s equations involving displacement current are given as follows:
  1. Gauss’s Law for Electric Fields: , where is the electric field, is the differential area, and is the charge enclosed.
  2. Gauss’s Law for Magnetic Fields: , where is the magnetic field, and is the differential area.
  3. Faraday’s Law of Electromagnetic Induction: , where is the electric field, is the differential length along the closed loop, is the differential magnetic field, and is the differential time.
  4. Ampere-Maxwell Law: , where is the magnetic field, is the differential length along the closed loop, is the current passing through the loop, is the permeability of free space, is the relative permeability of the medium, and is the electric flux.

Applications of Displacement Current

• Displacement current has various practical applications in different fields:
  • Electromagnetic Waves: Displacement current is essential for the generation and propagation of electromagnetic waves. These waves are extensively used in wireless communication, radar systems, and satellite communication.
  • Capacitors: Displacement current plays a role in the charging and discharging of capacitors. It affects the rate at which the voltage across a capacitor changes.
  • Inductors: Displacement current is involved in the behavior of inductors and affects their impedance. It is important in circuits with time-varying currents.
  • Electrical Insulation: Understanding displacement current is crucial in designing effective insulation systems to prevent electric breakdown and ensure safe and efficient electrical power transmission.
  • Medical Imaging: Displacement current principles are utilized in medical imaging techniques such as magnetic resonance imaging (MRI) and positron emission tomography (PET) for diagnosing and monitoring various medical conditions.
  • Electronics: Displacement current is considered in circuit analysis and designing electronic devices to ensure accurate performance and minimize electromagnetic interference.

11. Displacement Current - Magnetic Field Between Circular Parallel Plate Capacitors

• Displacement current is a concept introduced by Maxwell to explain the magnetic field between circular parallel plate capacitors.
• Derived from Ampere’s circuital law which relates the magnetic field around a closed loop to the current passing through the loop.
• Circular parallel plate capacitors have no actual current flowing between the plates, but a changing electric field exists.
• Displacement current is represented by the equation: .
• Displacement current contributes to the time-varying electric and magnetic fields.

12. Electric Field - Circular Parallel Plate Capacitors

• Circular parallel plate capacitors consist of two circular plates with opposite charges.
• Electric field between the plates is radial and given by .
• Electric field is directed from positive charges to negative charges.
• Magnitude of electric field decreases with distance from the center of the plates.
• Electric field is zero at the center of the circular plates.

13. Magnetic Field - Circular Parallel Plate Capacitors

• Due to displacement current, a magnetic field exists between circular parallel plate capacitors.
• Magnetic field forms concentric circles around the center of the plates.
• Magnitude of magnetic field inside the plates is given by , where is the displacement current and is the distance from the center of the plates.
• Magnetic field is stronger closer to the plates and weaker farther away.
• Direction of magnetic field depends on electric field direction and plate orientation.

14. Ampere’s Circuital Law

• Ampere’s circuital law relates magnetic field around a closed loop to the current passing through the loop.
• Line integral around the loop is equal to the product of current and permittivity of free space.
• Mathematically represented as , where is magnetic field, is differential length along the loop, and is current passing through the loop.
• Applies to cases with steady magnetic fields and no time-varying electric fields.

15. Derivation of Displacement Current

• Consider a circular parallel plate capacitor with changing electric field.
• Electric field between the plates is given by .
• Electric flux , where is the area of the plates.
• Differentiating electric flux with respect to time, we get .
• Substituting the value of electric field and differentiating, we obtain .
• Simplifying further, we get .
• This equation shows the existence of displacement current between circular parallel plate capacitors.

16. Role of Displacement Current in Maxwell’s Equations

• Displacement current is vital in Maxwell’s equations, which describe electric and magnetic fields.
• Inclusion of displacement current allows explanation of various electromagnetic phenomena, including electromagnetic wave propagation.
• Maxwell’s equations involving displacement current are:
  1. Gauss’s Law for Electric Fields: ![Gauss’s Law for Electric Fields](https://latex.codecogs.com/png.latex?%5Coiint%20E%20%5