UNIT 4 Uniform Plane Wave & Poynting Vector

 

What is a Uniform Plane Wave?

Uniform Plane Wave ek aisi electromagnetic wave hoti hai jisme:

• Electric field (E) aur Magnetic field (H) uniform hoti hain

• Wave ek fixed direction me travel karti hai

• Field ka variation sirf propagation direction me hota hai

👉 Example:

• Radio waves

• Microwave communication

• Free space propagation

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Wave Equation

Maxwell’s equations se wave equation derive hoti hai:

$$\nabla^2 E = \mu \epsilon \frac{\partial^2 E}{\partial t^2}$$ $$\nabla^2 H = \mu \epsilon \frac{\partial^2 H}{\partial t^2}$$

👉 Ye equation batati hai ki EM waves space aur time dono me propagate karti hain.

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Solution of Wave Equation

1. Dielectric Medium

• Conductivity ≈ 0

• Energy loss negligible

$$E = E_0 e^{-j\beta z}$$

👉 Mostly free space & insulating materials me hota hai.

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2. Conducting Medium

• Conductivity high hoti hai

• Wave amplitude gradually decay hoti hai

$$E = E_0 e^{-\alpha z} e^{-j\beta z}$$

👉 Metals me waves jaldi attenuate ho jaati hain.

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Free Space Propagation

Free space me:

• Conductivity = 0

• Relative permeability = 1

Important parameters:

• Intrinsic impedance:

$$\eta_0 = 377\ \Omega$$

• Velocity of propagation:

$$v = 3 \times 10^8\ m/s$$

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Surface Impedance

Surface impedance batata hai ki conductor EM wave ko kitna resist karta hai.

$$Z_s = \frac{E_t}{H_t}$$

👉 High conductivity → low penetration of wave.

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Depth of Penetration (Skin Depth)

Skin depth wo depth hai jahan wave ki amplitude 1/e ho jaati hai.

$$\delta = \frac{1}{\alpha}$$

👉 High frequency & good conductor → small skin depth

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Phase Velocity & Group Velocity

Phase Velocity

Speed jisse wave ka phase travel karta hai.

$$v_p = \frac{\omega}{\beta}$$

Group Velocity

Speed jisse energy or information travel karti hai.

$$v_g = \frac{d\omega}{d\beta}$$

👉 Communication systems me group velocity important hoti hai.

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Polarization of Uniform Plane Waves

Polarization batata hai ki Electric field ka orientation kya hai.

Types of Polarization

• Linear polarization

• Circular polarization

• Elliptical polarization

👉 Antenna design me polarization bahut important role play karti hai.

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Reflection of Plane Waves

1. Reflection by Perfect Conductor

• Incident wave completely reflect ho jaati hai

• Electric field surface par zero ho jaata hai

👉 Example:

• Metal surfaces

• Mirror-like reflection

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2. Reflection by Perfect Dielectric

Reflection depend karta hai:

• Angle of incidence

• Permittivity of medium

Normal Incidence

Wave perpendicular hit karti hai surface ko.

Oblique Incidence

Wave angle par incident hoti hai.

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Brewster Angle

Brewster angle wo angle hota hai jahan reflected wave completely polarized ho jaati hai.

$$\tan \theta_B = \sqrt{\frac{\epsilon_2}{\epsilon_1}}$$

👉 Is angle par parallel component reflection zero ho jaata hai.

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Poynting Vector and Flow of Power

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What is Poynting Vector?

Poynting vector batata hai:

Electromagnetic energy ka direction aur rate of flow

$$\vec{S} = \vec{E} \times \vec{H}$$

Unit: Watt/m²

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Poynting Theorem

Poynting theorem energy conservation ko explain karta hai.

Statement

Power supplied = Power stored + Power dissipated

👉 Ye theorem EM energy flow ko mathematically justify karta hai.

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Instantaneous Poynting Vector

Time ke kisi instant par power flow batata hai:

$$S(t) = E(t) \times H(t)$$

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Average Poynting Vector

AC waves ke liye average power:

$$S_{avg} = \frac{1}{2} Re(E \times H^*)$$

👉 Communication aur transmission analysis me use hota hai.

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Complex Poynting Vector

Complex form use hoti hai:

• Reactive power

• Stored energy analysis

$$S = \frac{1}{2} E \times H^*$$

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Conclusion

Uniform Plane Wave aur Poynting Vector concepts se:

• Wave propagation

• Reflection & transmission

• Energy flow

• Communication systems