Foundations — Electric field of point charge, dipole, ring, disk, line charge (Gauss's law)

1.8.5 · D1 · Physics › Electromagnetism › Electric field of point charge, dipole, ring, disk, line cha

Is page par assume kiya gaya hai ki tumne kuch bhi nahi dekha. Hum har woh symbol name karte hain jo parent topic note use karta hai, uske peeche ki picture draw karte hain, aur batate hain ki topic ko uski zaroorat kyun hai. Upar se neeche padho — har idea sirf upar waale ideas use karta hai.

1. "Charge" kya hai? Symbol $q$ (aur $Q$)

Do tarah ke dots ki picture banao: positive (inhe hum magenta mein draw karte hain) aur negative (inhe hum violet mein draw karte hain). Same type repel karte hain, opposite type attract karte hain. Yahi charge ki poori personality hai.

Topic ko isko kyun chahiye: charge source hai. Koi charge nahi, koi field nahi. Parent page par jo kuch bhi hai woh ya toh ek dot ($q$) se shuru hota hai ya bahut saare dots ki smear se jo $Q$ tak add hoti hai.

2. Distance $r$ aur "kitna door, kis taraf" wala arrow $\hat{r}$

Isse pehle ki hum batayein push kitni strong hai, humein chahiye kitna door hain hum aur kis direction mein.

Figure s01 dekho: magenta charge centre mein baitha hai, black point $P$ woh jagah hai jahan hum measure karte hain, dashed line ki length $r$ hai, aur chhota orange arrow uske saath chalta hai jo $\hat{r}$ hai — hamesha length 1, hamesha $P$ ki taraf bahar point karta hua.

"Kitna door" ko "kis taraf" se alag kyun rakhte hain: strength aur direction do alag sawaal hain. $r$ jawaab deta hai kitna strong (through $1/r^2$, agla section). $\hat{r}$ jawaab deta hai push kis taraf point karti hai. Inhe alag rakhna hi baad mein yeh kehne deta hai "ye directions cancel hoti hain, woh add hoti hain."

3. Vectors aur arrow-sum (superposition)

Do arrows ko tip-to-tail add kiya jaata hai: doosre arrow ko slide karo taaki uski tail pehle arrow ki tip par baithe; sum woh arrow hai jo bilkul pehli tail se bilkul aakhri tip tak jaata hai.

Figure s02 mein do patli arrows ek hi point par do alag charges ki pushes hain; mothi navy arrow unka tip-to-tail sum hai. Notice karo upar-neeche ke parts cancel ho jaate hain aur sirf sideways part bachta hai — yahi cancellation ring aur dipole results ke peeche ki poori trick hai.

Topic ko isko kyun chahiye: ek real object mein lakho charges hote hain. Jo field yeh banata hai woh har ek ka arrow-sum hai. Us rule ka ek naam hai.

4. Field $\vec{E}$ khud — arrows ka matlab kya hai

Poore room mein chhote arrows ki ek grid ki picture banao, har point par ek, ek positive dot ko batate hue ki use kis taraf dhakela jayega aur kitna zor se. Positive charge ke paas arrows *door point karte hain; negative charge ke paas yeh uski taraf point karte hain (dekho Flux and field lines).

"Per unit charge" kyun: hum chahte hain ki field space ko khud describe kare, koi bhi test charge ko nahi. Test charge ko divide karne se $\vec{E}$ sirf source ki property ban jaati hai.

5. Constants $k$, ${\epsilon }_0$, aur $\pi$

Formulas ko ek aisi number chahiye jo "charge ki matra aur distance" ko "actual push" mein convert kare. Parent page par har jagah ek hi constant ki do spellings aati hain.

Ek cheez ke liye do naam kyun: $k$ point-charge formulas ($kq/r^2$) ke liye compact hai; $\frac{1}{4\pi {\epsilon }_0}$ wala form Gauss's law ke liye natural hai kyunki wahaan ka $4\pi$ literally ek sphere ki surface hai. Same number, occasion ke hisaab se alag dress.

6. $1/r^2$ falloff — "sphere par failna" kyun

Ek fixed number of field arrows ek charge se nikalte hain aur fan out ho jaate hain. Distance $r$ par woh ek sphere ki surface par faile hue hain. Radius $r$ ke sphere ki surface ka area hota hai

$A=4\pi r^2.$

Toh arrows ki density — jo field strength hai — hai (arrows) $÷$ (area) $\propto 1/r^2$. Figure s03 mein wahi lines ek choti nazdiki sphere ko densely aur ek badi door sphere ko sparsely chedhti hain.

Topic ko isko kyun chahiye: yeh ek geometric fact ($1/r^2$) sab kuch ki seed hai — Coulomb's law, aur baad mein yeh fact ki ek dipole tezi se girta hai ($1/r^3$) jabki ek line dheere ($1/r$).

7. Density symbols: $\lambda$, $\sigma$ — "charge kitna packed hai?"

Jab charge kisi shape par faila hota hai, hum batate hain ki yeh kitna thickly packed hai.

Ek magenta wire ki picture banao: uski $\Delta \ell$ length kato, aur usme charge $\lambda \Delta \ell$ hai. Ek sheet ki picture banao: area $\Delta A$ ka ek patch kato, usme charge $\sigma \Delta A$ hai.

Topic ko inki kyun zaroorat hai: parent page mein line, disk, aur sheet ke charge spread out hain, isliye hum sirf $q$ nahi likh sakte — hum "density × chhote piece ka size" likhte hain har $dq$ paane ke liye (agla section).

8. Little-piece symbols: $dq$, $ds$, $dA$, aur integral $\int$

Ek faile hue charge ko arrow-sum karne ke liye hum ise aise pieces mein chop karte hain ki ek piece ek point ki tarah act kare.

Har piece banana: $\text{line: }dq=\lambda ds,\text{ring of radius }s:dq=\sigma (2\pi s)ds.$ "$2\pi sds$" kyun: radius $s$ aur width $ds$ ki ek patli ring ki circumference $2\pi s$ hai, isliye uska area (circumference)×(width) $=2\pi sds$ hai, aur uska charge $\sigma$ times that hai.

Topic ko isko kyun chahiye: ring, disk, aur line results kuch nahi hain sirf is chop-and-add ke, ek $\int$ pahne hue.

9. Angle $\theta$, $\cos \theta$, aur "kaun se components survive karte hain"

Jab arrows tilted directions mein point karte hain, hum sirf woh part rakhte hain jo cancel nahi hota.

Figure s04 mein, ek ring element se ek tilted field arrow ki length $dE$ hai. Axis ke along (woh direction jo survive karta hai) wala part $dE\cos \theta$ hai, jahaan $\cos \theta =\frac{x}{\sqrt{R^2+x^2}}$ kyunki adjacent side axial distance $x$ hai aur hypotenuse poori distance $\sqrt{R^2+x^2}$ hai.

Topic ko isko kyun chahiye: ring ke axis par, har element ke sideways parts symmetry se cancel ho jaate hain; sirf axial part $dE\cos \theta$ bachta hai. $\cos \theta$ se multiply karna hi woh tarika hai jisse hum doomed component ko fenk dete hain.

10. Flux, aur closed-loop integral $\oint$ (Gauss machinery)

Parent Gauss's law ko shortcut ki tarah use karta hai. Pehle do symbols samajhne zaroori hain.

Topic ko isko kyun chahiye: jab koi shape itni symmetric ho (sphere, cylinder, sheet), toh $\vec{E}$ constant aur perpendicular hota hai ek chalaaki se chosen surface par, isliye daraaona $\oint$ (field)×(area) mein collapse ho jaata hai — ek turant answer. Dekho bhi Coulomb's law aur Flux and field lines.

Prerequisite map

Equipment checklist

Khud se test karo — parent topic padhne se pehle har ek ka jawaab dena chahiye.