Foundations — Poynting vector — energy flux in EM waves

1.8.36 · D1 · Physics › Electromagnetism › Poynting vector — energy flux in EM waves

Pehle aap parent note Poynting vector padh sako, uske liye tumhe har woh symbol khud se samajhna hoga jo woh tumhare saamne fenkta hai. Neeche, har piece ko zero se build kiya gaya hai: saral words → ek picture → kyun is topic ko yeh chahiye. Yeh is order mein hain ki har ek pichle pe tikta hai.

0. Woh picture jiske andar sab kuch baith ta hai

Sab kuch ordinary 3D space mein hota hai. Ek vector ek arrow hai: uski ek length hoti hai (kitna bada) aur ek direction (kis taraf). Hum vectors ko ek chhote hat-arrow ke saath likhte hain, jaise $\vec{E}$. Ek plain letter jaise $E$ (bina arrow ke) sirf us arrow ki length ka matlab hai — bina direction ke ek akela number.

1. Electric field $\vec{E}$

Yeh topic kyun chahiye: wave ki energy ka aadha hissa $\vec{E}$ mein rehta hai. Energy density term $\frac{1}{2}{\epsilon }_0{E}^2$ aur poora Poynting vector dono yahan se shuru hote hain.

2. Magnetic field $\vec{B}$

Yeh topic kyun chahiye: wave ki energy ka doosra aadha $\vec{B}$ mein rehta hai, aur $\vec{S}$ literally $\vec{E}$ aur $\vec{B}$ ka combination hai. Ek light wave mein $\vec{E}$ aur $\vec{B}$ hamesha right angles par hote hain aur ek saath locked rehte hain (aap dekhoge $E=cB$).

3. Cross product $\vec{E}\times \vec{B}$

Yeh is topic mein sabse important operation hai, isliye hum ise carefully build karte hain.

Figure dekho. $\sin \theta$ kyun? Yeh measure karta hai ki do arrows kitne "alag" hain:

• Agar woh same direction mein point karen ($\theta =0$), $\sin 0=0$ — result zero hai. Do parallel fields koi propagating wave carry nahi karte, isliye koi energy flow nahi.
• Agar woh perpendicular hon ($\theta =90^{\circ }$), $\sin 90^{\circ }=1$ — result jitna ho sake utna bada hota hai.

Light wave mein $\vec{E}\perp \vec{B}$, isliye $\theta =90^{\circ }$ aur length khoobsurat tarike se simplify ho jati hai: $|\vec{E}\times \vec{B}|=EB\sin 90^{\circ }=EB.$ Yahi exact reason hai ki parent note likh sakta hai $S=EB/{\mu }_0$.

4. Dot product aur surface element $d\vec{A}$

Yeh topic kyun chahiye: ek surface ko cross karne wali total power paane ke liye, parent likhta hai $P=\oint \vec{S}\cdot d\vec{A}$. Dot product sirf woh energy flow pick karta hai jo actually har patch ko pierce karta hai (energy jo surface ke saath sideways skim karti hai, kuch cross nahi karti). $\oint$ symbol ka matlab sirf "poori closed surface par add karo."

5. Constants ${\epsilon }_0$, ${\mu }_0$, aur speed $c$

Yeh topic kyun chahiye: $\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B}$ mein ${\mu }_0$ hai. Identity $\frac{1}{{\mu }_{0}c}={\epsilon }_{0}c$ (jo tumhe $\frac{1}{{\mu }_{0}c}$ ke top aur bottom ko ${\epsilon }_{0}c$ se multiply karke aur $c^2=1/({\mu }_0{\epsilon }_0)$ use karke milti hai) wahi hai jo $S=E^2/({\mu }_{0}c)$ ko $S={\epsilon }_{0}c{E}^2$ mein badal deti hai. Dekho EM wave equation ki $c$ kahan se aata hai.

6. Energy density $u$

Yeh topic kyun chahiye: derivation yeh track karke shuru hoti hai ki $u$ time ke saath kaise change hoti hai aur pucha jata hai "energy kahan gayi?" Woh jawab hi $\vec{S}$ hai. Prerequisite: Energy density of electric and magnetic fields.

7. Rate of change $\frac{\partial u}{\partial t}$ aur divergence $\nabla \cdot$

Yeh topic kyun chahiye: Poynting's theorem ek bookkeeping line hai, $\frac{\partial u}{\partial t}+\nabla \cdot \vec{S}=-\vec{J}\cdot \vec{E},$ ise padho "(energy pile ho rahi hai) + (energy bahar flow ho rahi hai) = (minus work done on charges)." $\nabla \cdot$ exactly woh tool hai "bahar flow ho rahi hai" ke liye.

8. Curl $\nabla \times$, current density $\vec{J}$, aur Maxwell's laws

Yeh topic kyun chahiye: derivation Maxwell's equations use karke fields ke time-derivatives ko curls se swap karti hai — woh akele laws jo batate hain ki $\vec{E}$ aur $\vec{B}$ kaise evolve karte hain. Phir Vector calculus identities ki ek line, $\nabla \cdot (\vec{E}\times \vec{B})=\vec{B}\cdot (\nabla \times \vec{E})-\vec{E}\cdot (\nabla \times \vec{B}),$ do curl terms ko $\vec{E}\times \vec{B}$ ki single divergence mein package karti hai — aur $\vec{S}$ bahar pop hota hai.

9. Time-average $⟨⟩$ aur intensity $I$

Yeh topic kyun chahiye: $E_0$ peak field amplitude hai; measurements $I$ dete hain. Factor $\frac{1}{2}$ (woh famous forgotten hero) poora $⟨{\cos }^2⟩=\frac{1}{2}$ se aata hai. Iske baare mein aur Intensity and amplitude of waves mein.

Prerequisite map

Equipment checklist

Test karo khud ko — parent note kholne se pehle har ek ka jawab dena aana chahiye.