UNIT–3: Time Varying Fields and Maxwell’s Equations

 

What are Time Varying Fields?

Jab electric field (E) aur magnetic field (B) time ke sath change hoti hain, to unhe time varying fields kehte hain.

👉 Ye fields mainly yahan use hoti hain:

• AC circuits

• Transformers

• Electric motors & generators

• Antennas

• Wireless communication

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Equation of Continuity for Time Varying Fields

Equation of continuity charge conservation law ko represent karti hai.

Statement

Electric charge neither be created nor destroyed, it can only flow.

Mathematical Expression

$$\nabla \cdot J = -\frac{\partial \rho}{\partial t}$$

Explanation

• Agar current kisi region se bahar ja raha hai

• To wahan charge density kam ho jaati hai

👉 Ye equation time-varying current ke liye valid hoti hai.

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Maxwell’s Equations – Introduction

James Clerk Maxwell ne 4 equations di jinhone electric aur magnetic fields ko unify kiya.
Inhi equations par modern electrical & communication systems based hain.

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Maxwell’s Equations (Differential Form)

1. Gauss’s Law for Electricity

$$\nabla \cdot D = \rho_v$$

👉 Electric flux density ka divergence = charge density

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2. Gauss’s Law for Magnetism

$$\nabla \cdot B = 0$$

👉 Magnetic monopole exist nahi karta

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3. Faraday’s Law

$$\nabla \times E = -\frac{\partial B}{\partial t}$$

👉 Time-varying magnetic field → electric field produce karti hai

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4. Ampere–Maxwell Law

$$\nabla \times H = J + \frac{\partial D}{\partial t}$$

👉 Yahan displacement current introduce hota hai

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Maxwell’s Equations (Integral Form)

• Electric Gauss Law

$$\oint D \cdot dS = Q$$

• Magnetic Gauss Law

$$\oint B \cdot dS = 0$$

• Faraday’s Law

$$\oint E \cdot dl = -\frac{d}{dt} \int B \cdot dS$$

• Ampere–Maxwell Law

$$\oint H \cdot dl = I + \frac{d}{dt} \int D \cdot dS$$

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Boundary Conditions in Electromagnetic Fields

Jab EM fields do alag media ke boundary par hoti hain, to unka behavior change hota hai.

Electric Field

• Tangential E → continuous

• Normal D → surface charge par depend

Magnetic Field

• Normal B → continuous

• Tangential H → surface current par depend

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Faraday’s Law of Electromagnetic Induction

Statement

Whenever magnetic flux linked with a conductor changes, an EMF is induced.

Formula

$$e = -\frac{d\Phi}{dt}$$

👉 Negative sign Lenz’s Law ko show karta hai.

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Transformer EMF

• Conductor stationary hota hai

• EMF magnetic flux ke time change se induce hoti hai

Example

• Power transformer

• Induction coil

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Motional EMF

• Conductor magnetic field me move karta hai

Formula

$$e = B l v$$

Example

• Electric generator

• DC motor

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Time Harmonic Fields

Jab electric aur magnetic fields sinusoidal form me hoti hain:

$$E = E_0 \sin \omega t$$

👉 Inhe time harmonic fields kehte hain.
AC systems aur communication me bahut use hoti hain.

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Electromagnetic Potentials

Scalar Potential

$$E = -\nabla V$$

Vector Potential

$$B = \nabla \times A$$

👉 Potentials se EM field calculation easy ho jaati hai.

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Relation Between Circuit Theory and Field Theory

Circuit Theory	Field Theory
Lumped parameters	Distributed parameters
KVL, KCL	Maxwell’s equations
Low frequency	High frequency
Simple analysis	Accurate analysis

👉 Circuit theory, field theory ka special case hai.

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Conclusion

Time Varying Fields aur Maxwell’s Equations Electromagnetic Theory ka backbone hain.
Ye concepts samajh aane ke baad AC machines, communication systems, antennas sab clear ho jaate hain.