Electromagnetic Waves | Class 12 Physics Notes

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.

📊 Diagram: See figure_2, figure_3, figure_4: (a) A parallel plate capacitor C, as part of a circuit through which a time dependent current i(t) flows, with a loop of radius r to determine magnetic field at point P; (b) a pot-shaped surface passing through the interior between the capacitor plates with the loop as its rim; (c) a tiffin-shaped surface with the circular loop as its rim and a flat circular bottom S between the capacitor plates. Arrows show uniform electric field between the capacitor plates. See figure_5 and figure_6: (a) The electric and magnetic fields E and B between the capacitor plates at point M; (b) cross-sectional view of the fields.

🧪 Activity: Measuring the magnetic field at a point between capacitor plates to verify the presence of displacement current.

🔗 Connection: This section leads to Maxwell's equations in vacuum, summarizing the fundamental laws of electromagnetism including displacement current.