Maxwell's Equation and Displacement Current: Comprehensive NEET Physics Notes

1. Maxwell's Equation and Displacement Current

1.1 Introduction to Displacement Current

In Chapter 4, we learned that an electric current produces a magnetic field and that a changing magnetic field induces an electric field. Maxwell proposed that a time-varying electric field should also give rise to a magnetic field, similar to how a magnetic field changing over time generates an electric field. This idea led to the concept of displacement current, which addressed an inconsistency in Ampere’s circuital law.

Key Idea

Maxwell noticed that when applying Ampere’s circuital law to a charging capacitor, the law seemed inconsistent. He introduced the concept of displacement current to resolve this inconsistency and ensure the law's applicability in all situations.

1.2 Understanding Displacement Current

The displacement current is not an actual flow of charge but arises due to a time-varying electric field. To understand this, let’s consider the process of charging a capacitor:

1. When a capacitor is connected to a time-dependent current $i(t)$, a changing electric field is established between the plates.
2. According to Ampere's law, the magnetic field at a point outside a capacitor should be determined by the conduction current $i(t)$ flowing in the wires. However, if we choose a surface that passes between the capacitor plates (where there is no conduction current), Ampere's law seems to fail.

Calculation of Displacement Current

The inconsistency is resolved by introducing the displacement current $i_d$:

• Consider a parallel plate capacitor with charge $Q$ on the plates.
• The electric field between the plates is given by: $E=\frac{Q}{A{ϵ}_0}$ where $A$ is the area of the capacitor plates and ${ϵ}_0$ is the permittivity of free space.
• The electric flux ${\Phi }_E$ through the surface between the plates is: ${\Phi }_E=E\cdot A=\frac{Q}{{ϵ}_0}$
• Differentiating with respect to time: $\frac{d{\Phi }_E}{dt}=\frac{1}{{ϵ}_0}\frac{dQ}{dt}$
• Since $i=\frac{dQ}{dt}$, we have: $i_d={ϵ}_0\frac{d{\Phi }_E}{dt}$

This displacement current $i_d$ behaves like a real current in generating a magnetic field, even though no actual charges flow through the space between the capacitor plates.

Enhanced Visual Aid

![Include Diagram Here] A diagram showing a parallel plate capacitor with the electric field lines between the plates and the concept of displacement current would be beneficial. The illustration should depict both the conduction current and the displacement current to highlight the difference.

1.3 Generalized Ampere's Law (Ampere-Maxwell Law)

Maxwell’s modification of Ampere's law is expressed as:

$\oint \vec{B}\cdot d\vec{l}={\mu }_0\left(i_c+{ϵ}_0\frac{d{\Phi }_E}{dt}\right)$

Where:

• ${\mu }_0$: Permeability of free space
• $i_c$: Conduction current
• $i_d={ϵ}_0\frac{d{\Phi }_E}{dt}$: Displacement current

This equation, known as the Ampere-Maxwell Law, unifies the concept of electric currents and displacement currents as sources of the magnetic field.

Real-life Application

The concept of displacement current is crucial in understanding how electromagnetic waves propagate through space, which is the foundation of wireless communication technologies like radio, TV, and mobile communication.

Did You Know?

Maxwell's equations predicted the existence of electromagnetic waves traveling at the speed of light, leading to the realization that light itself is an electromagnetic wave.

Common Misconception

Many students mistakenly believe that displacement current refers to actual moving charges. In reality, it represents the effect of a changing electric field, not a physical current.

Quick Recap

• Displacement Current arises from a changing electric field, even in the absence of a physical flow of charge.
• Ampere-Maxwell Law combines conduction current and displacement current to give a complete picture of the sources of the magnetic field.
• The displacement current ensures the continuity of magnetic field lines, even in regions without conduction current.

NEET Exam Strategy

• Focus on the derivation and application of displacement current in Ampere’s circuital law.
• Understand how Maxwell's correction to Ampere's law leads to the concept of electromagnetic waves.
• Practice applying the concept of displacement current to different scenarios, as NEET often includes questions requiring a conceptual understanding of this topic.

Practice Questions

Question 1

A parallel plate capacitor with a plate area of 0.1 m2^22 and plate separation of 0.01 m is being charged such that the electric field between the plates changes at a rate of 2×1062 \times 10^62×106 V/m/s. Calculate the displacement current between the plates.

Solution:

Given:

• Plate area, $A=0.1{\text{m}}^2$
• Rate of change of electric field, $\frac{dE}{dt}=2\times 10^6\text{V/m/s}$
• Displacement current density, $J_d={ϵ}_0\frac{dE}{dt}$

Using $i_d=J_d\cdot A$: $i_d={ϵ}_0\frac{dE}{dt}\cdot A=8.85\times 10^{-12}\times 2\times 10^6\times 0.1=1.77\times 10^{-6}\text{A}$

Question 2

If the electric flux through a surface changes at a rate of $5\times 10^4\text{Vm}$ per second, what is the displacement current?

Solution:

Using: $i_d={ϵ}_0\frac{d{\Phi }_E}{dt}$ $i_d=8.85\times 10^{-12}\times 5\times 10^4=$