MP Board · Class 12 · Physics · Electromagnetic WavesWhat is displacement current? Why was it introduced by James Clerk Maxwell? State Maxwell's modified Ampere's circuital law and derive its mathematical expression.

Introduction to Displacement Current\nDisplacement current is a quantity that appears in Maxwell's equations and is defined in terms of the rate of change of electric displacement field or electric flux. It was conceptualized by the great physicist James Clerk Maxwell to resolve a fundamental inconsistency in Ampere's circuital law.

Need for Displacement Current\nAmpere's circuital law states that the line integral of magnetic field $\vec{B}$ around any closed loop is equal to $\mu_0$ times the total current $I$ passing through the surface bounded by the loop:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I$$ \nHowever, consider a parallel plate capacitor being charged by a direct current source. If we consider two different surfaces (S1 and S2) bounded by the same closed loop — where S1 is a flat surface passing right through the wire carrying conduction current $I$, and S2 is a balloon-shaped surface passing through the space between the capacitor plates where no conduction current exists — applying Ampere's law gives inconsistent results:

• For surface S1: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$
• For surface S2: $\oint \vec{B} \cdot d\vec{l} = 0$ \nSince line integrals cannot yield different values for the same closed loop, Maxwell concluded that Ampere's law was incomplete for non-steady currents.

Maxwell's Correction and Displacement Current\nMaxwell realized that although there is no conduction current in the region between the capacitor plates, there is a changing electric field due to the accumulation of charge on the plates. The electric flux $\Phi_E$ linked between the plates changes with time.

\nHe defined the displacement current $I_d$ as: $$I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$$\nwhere $\varepsilon_0$ is the permittivity of free space and $\frac{d\Phi_E}{dt}$ is the rate of change of electric flux.

Maxwell's Modified Ampere's Circuital Law\nTo make Ampere's law universally consistent, Maxwell added the displacement current to the conduction current. The total current $I_{total}$ is the sum of conduction current ($I_c$) and displacement current ($I_d$):

$$I_{total} = I_c + I_d = I_c + \varepsilon_0 \frac{d\Phi_E}{dt}$| \nThus, the modified Ampere-Maxwell law is expressed as: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_c + \varepsilon_0 \frac{d\Phi_E}{dt} \right)$| \nThis crucial addition not only resolved the mathematical paradox in electromagnetism but also led to the theoretical prediction of electromagnetic waves.