Poynting vector — energy flux in EM waves

1.8.36 · Physics › Electromagnetism

Poynting vector HAI kya?

Cross product kyun? Energy flow ko ek direction chahiye, aur woh dono $\vec{E}$ aur $\vec{B}$ ke perpendicular honi chahiye (wahin wave jaati hai). Cross product woh natural object hai jo dono ke perpendicular hota hai aur zero ho jaata hai jab fields parallel hon (koi propagating wave nahi).

Ise first principles se KAISE derive karein

Hume ek bookkeeping equation chahiye: stored field energy ke change ki rate + outflow = charges par fields dwara kiya gaya kaam. Maxwell's equations aur energy density se shuru karo.

Step 2 — $u$ ki time derivative lo: $\frac{\partial u}{\partial t}={\epsilon }_0\vec{E}\cdot \frac{\partial \vec{E}}{\partial t}+\frac{1}{{\mu }_0}\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}$ Kyun? Hum track karte hain ki stored energy time mein kaise change hoti hai taaki hum dhundh sakein woh kahan gayi.

Step 3 — Maxwell's curl equations substitute karo (free space + currents): $\nabla \times \vec{B}={\mu }_0\vec{J}+{\mu }_0{\epsilon }_0\frac{\partial \vec{E}}{\partial t}\Rightarrow {\epsilon }_0\frac{\partial \vec{E}}{\partial t}=\frac{1}{{\mu }_0}\nabla \times \vec{B}-\vec{J}$ $\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\Rightarrow \frac{\partial \vec{B}}{\partial t}=-\nabla \times \vec{E}$ Kyun? Maxwell's equations wahi ek laws hain jo humein batate hain ki fields kaise evolve hoti hain — inhe substitute karne se "energy change ki rate" currents aur field geometry ke baare mein kuch ban jaata hai.

Step 4 — Plug in karo: $\frac{\partial u}{\partial t}=\vec{E}\cdot \left(\frac{1}{{\mu }_0}\nabla \times \vec{B}-\vec{J}\right)-\frac{1}{{\mu }_0}\vec{B}\cdot (\nabla \times \vec{E})$

Step 5 — Vector identity use karo $\nabla \cdot (\vec{E}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{E})-\vec{E}\cdot (\nabla \times \vec{B})$: $\frac{1}{{\mu }_0}[\vec{E}\cdot (\nabla \times \vec{B})-\vec{B}\cdot (\nabla \times \vec{E})]=-\frac{1}{{\mu }_0}\nabla \cdot (\vec{E}\times \vec{B})=-\nabla \cdot \vec{S}$ Kyun? Yeh identity exactly wahi cheez hai jo hume dono curl terms ko ek single vector ke divergence mein package karne deti hai — woh vector $\vec{S}$ hai.

Ek plane EM wave ke liye specialize karna

Kyunki $\vec{E}\perp \vec{B}$, $|\vec{E}\times \vec{B}|=EB$, isliye $S=\frac{EB}{{\mu }_0}=\frac{E(E/c)}{{\mu }_0}=\frac{E^2}{{\mu }_{0}c}={\epsilon }_{0}c{E}^2$ $B=E/c$ kyun? Maxwell's wave solution se field amplitudes wave speed dwara lock hote hain; aur $\frac{1}{{\mu }_{0}c}={\epsilon }_{0}c$ using $c^2=1/({\mu }_0{\epsilon }_0)$.

Intensity = time-averaged S. $E=E_0\cos (kx-\omega t)$ ke saath, $⟨{\cos }^2⟩=\frac{1}{2}$:

Worked examples

Common mistakes (steel-manned)

Yaad rakhne ke liye useful relations

Flashcards

Connections

• Maxwell's equations — derivation mein use ki gayi curl substitutions ka source.
• EM wave equation — $E=cB$ aur $c^2=1/{\mu }_0{\epsilon }_0$ deta hai.
• Energy density of electric and magnetic fields — woh $u$ jise humne differentiate kiya.
• Radiation pressure — $\vec{S}$ ka momentum analog ($\vec{g}=\vec{S}/c^2$).
• Vector calculus identities — $\nabla \cdot (\vec{E}\times \vec{B})$ identity.
• Intensity and amplitude of waves — time-averaging step.

Concept Map