Coulomb's law — force, comparison with gravity: Visual walkthrough

1.8.2 · D2 · Physics › Electromagnetism › Coulomb's law — force, comparison with gravity

Hum sirf woh ideas use karenge jo ek 12-saal-ka bachcha pehle se jaanta hai: ek dot, ek distance, ek arrow, ek ball ki surface, aur yeh fact ki "same cheez ko zyada area par failaane se woh patli ho jaati hai."

Step 1 — Ek charge ek dot hai jo "har direction mein bahar nikalti hai"

KYA HAI. Ek chhoti si charged speck draw karo — ek point charge. Iske paas jo charge hai use hum $q_{1}$ kehte hain (padho: "q-one"). Yeh chhota dot sizeless hai; saari charge ek hi point par hai.

YAH SE KYUN SHURU KAREIN. Pehle do charges ek doosre ko push kar sakein, humein picture karni hogi ki ek charge apne aas-paas ke space ke saath kya karta hai. Coulomb's law sirf inhi idealized dots ke liye exact hai (parent ke last mistake mein dekho), isliye hum honestly ek dot se shuru karte hain.

PICTURE. Dot se arrows sabhi directions mein barabar failte hain — upar, neeche, left, right, diagonal. Koi direction special nahi hai. Yeh "har direction mein samaanat" poori derivation ka beej hai.

Figure — Coulomb's law — force, comparison with gravity

Step 2 — Doosra charge $r$ distance par rakho: ab ek force hai

KYA HAI. Ek doosra dot rakho, charge $q_{2}$ , $q_{1}$ se $r$ distance par. Do dots ko join karne wali seedhi line ab ek hi special direction hai. Us line ke saath har charge ek push ya pull feel karta hai — ek force, jo hum ek single arrow ki tarah draw karte hain.

KYUN. Force ke liye do cheezein chahiye; ek akela charge kuch feel nahi karta. Jis moment ek partner aata hai, ek direction (joining line) aur ek strength (kitna zyada) paida hoti hai. Hamara poora kaam woh strength dhundhna hai.

PICTURE. Do dots, unke beech ek dashed line jis par $r$ likha hai, aur $q_{2}$ par ek red force arrow jo us line ke saath point karta hai.

Ab tak ke symbols reveal karo:

q_{1}

hai

pehle (source) dot par charge.

q_{2}

hai

doosre dot par charge, jis par hum force dekh rahe hain.

r

hai

do dots ke beech ki straight-line distance.

Step 3 — Strength $1/ r^{2}$ kyun fall off karti hai (spreading ball)

KYA HAI. $q_{1}$ ke influence ko ek fixed amount ka "spray paint" samjho jo dot se bahar jaata hai aur baahir ki taraf failta hai. $r$ distance par yeh sab radius $r$ wali ek ball (sphere) ki surface par phaila hua hai. Us ball ki surface area $4 π r^{2}$ hai. Toh $r$ distance par paint ki thickness hai

$yahan kitna strong influence density = \frac{total influence fixed, kabhi kho nahi jaata}{area jis par failta hai 4 π r ^{2}} .$

$1/ r$ ya kuch aur nahi, $1/ r^{2}$ KYUN? Kyunki space three-dimensional hai, aur ek badhti sphere ki surface radius ke square ki tarah badhti hai. Hum area use karte hain — circumference nahi, volume nahi — kyunki influence conserved hai aur woh surface hi hai jis se guzar ke woh distribute hoti hai. $r$ double karo → ball ki skin $4 \times$ badi ho jaati hai → wahi paint $4 \times$ patli ho jaati hai. Yahi inverse-square law ki poori origin hai.

PICTURE. Do concentric spheres, radius $r$ aur radius $2 r$ , baahri wali char guna area wali, utni hi arrows ek aise patch ko chedhti hain jo char guna phaila hua hai.

$A = 4 π r^{2}$ par term-by-term:

$A = fixed number ek poori sphere ke liye 4 π radius squared = isliye inverse-SQUARE r^{2}$

Ab tak: $strength \propto \frac{1}{r ^{2}}$ .

Step 4 — Kisi bhi charge ke double hone par strength kyun double hoti hai ( $\propto q_{1} q_{2}$ )

KYA HAI. Ab poochho ki force kitni charge har dot carry karta hai us par depend karti hai. $q_{1}$ ko do barabar hisson mein split karo jo ek doosre ke upar baithe hain. Har aadha apni force ke saath $q_{2}$ ko kheenchta hai; dono forces same direction mein point karti hain aur simply add ho jaati hain. Toh charge double → force double. Yahi argument $q_{2}$ par bhi apply hota hai.

Multiply kyun, add kyun nahi, do charges ko? Kyunki $q_{1}$ double karne se force double hoti hai aur $q_{2}$ double karne se bhi force double hoti hai. Ek quantity jo kisi bhi do inputs ke double hone par double hoti hai woh unke product ke proportional honi chahiye — yahi product karta hai. Isliye $strength \propto q_{1} q_{2}$ , kabhi $q_{1} + q_{2}$ nahi.

PICTURE. Left mein, $q_{1}$ ek dot ki tarah $q_{2}$ par ek force arrow ke saath. Right mein, $q_{1}$ do stacked half-dots ki tarah dikhaya gaya hai, do arrows ek double-lamba arrow mein add hote hue.

Steps 3 aur 4 combine karo:

$strength \propto \frac{q _{1} q _{2}}{r ^{2}} .$

Step 5 — "Proportional to" ko equation mein badlo: constant $k$

KYA HAI. "Proportional to" ( $\propto$ ) ka matlab hai "ek fixed multiplying number tak equal." Nature ko woh number supply karna hoga taaki units aur measured size sahi nikle. Use $k$ kehte hain. Tab

$F = Coulomb constant size aur units set karta hai k woh shape jo humne derive ki \frac{q _{1} q _{2}}{r ^{2}} .$

Constant ki zaroorat hi kyun hai? Spreading argument shape fix karta hai ( $q_{1} q_{2} / r^{2}$ ) lekin scale nahi. Experiment scale ek baar measure karta hai aur use pack karta hai $k = \frac{1}{4 π ε _{0}} \approx 8.99 \times 1 0^{9} N\cdotm^{2} / C^{2},$ mein, jahan $ε_{0}$ (permittivity of free space) record karta hai ki empty space electric influence carry karne ke liye kitna "tayyar" hai.

PICTURE. $k$ likha ek dial jo abstract shape $q_{1} q_{2} / r^{2}$ ko real number of newtons mein badalta hai.

k

hai

constant

1/ (4 π ε_{0}) \approx 8.99 \times 1 0^{9}

N·m²/C² jo shape ko real newtons mein badalta hai.

Scalar form mein absolute value kyun?

taaki magnitude hamesha positive rahe; direction alag se handle ki jaati hai.

Step 6 — Sign decide karta hai push ya pull (chaaon cases)

KYA HAI. Ek charge positive ya negative ho sakta hai. Do signed charges multiply karo. Agar product $q_{1} q_{2}$ positive hai (dono + ya dono −), charges repel karte hain — arrows ek doosre se door point karte hain. Agar product negative hai (ek +, ek −), woh attract karte hain — arrows ek doosre ki taraf point karte hain.

Product ka sign kyun kaafi hai. Sirf chaar sign combinations hain, aur woh do outcomes mein collapse ho jaate hain:

$q_{1}$	$q_{2}$	product $q_{1} q_{2}$	result
$+$	$+$	$+$	repel
$-$	$-$	$+$	repel
$+$	$-$	$-$	attract
$-$	$+$	$-$	attract

"Like repels, unlike attracts" bas yahi hai "product positive → apart, product negative → together." Vector form yeh automatically karta hai:

$F_{21} = k \frac{q _{1} q _{2}}{r ^{2}} \overset{r}{^}_{21},$

jahan $\overset{r}{^}_{21}$ length-1 arrow hai (ek unit vector) jo 1 se 2 ki taraf point karta hai. Agar $q_{1} q_{2} > 0$ toh poori cheez $\overset{r}{^}_{21}$ ke saath point karti hai (1 se door → repel); agar $q_{1} q_{2} < 0$ toh yeh flip ho jaati hai (1 ki taraf → attract).

PICTURE. Chaar chhote panels, table ki har row ke liye ek, har ek mein do dots aur unke force arrows.

Step 7 — Degenerate aur limiting cases (kabhi surprise mat lo)

KYA AUR KYUN — har edge chalo:

$r \to 0$ (charges merge ho jaate hain). $F = k ∣ q_{1} q_{2} ∣/ r^{2} \to \infty$ . Formula blow up ho jaata hai kyunki do point charges ko arbitrarily close push kiya ja sakta hai. Real objects ka size aur structure hota hai, isliye yeh infinity signal hai ki point-charge idealisation apply honi band ho gayi hai.
$r \to \infty$ (bahut door). $F \to 0$ , lekin dheere — $1/ r^{2}$ ki tarah, kabhi abruptly nahi. Pahunch kabhi truly khatam nahi hoti; sirf fade hoti hai.
$q_{1} = 0$ ya $q_{2} = 0$ . $F = 0$ . Ek neutral partner electric force feel nahi karta aur exert bhi nahi karta — yahi reason hai ki bade neutral bodies kamzor gravity ko jeetne dete hain.
Extended (non-point) charges. Formula as written sirf points ke liye apply hota hai (ya perfect spheres ke liye, jinhein all-charge-at-centre treat kiya jaata hai). Ek blob ke liye use tiny points mein chop karo aur har contribution add karo — phir se Superposition Principle.

PICTURE. $F$ vs $r$ ka ek single graph: $r = 0$ ke paas ek steep cliff, bade $r$ ki taraf ek lamba patla tail, dono limits annotate kiye hue.

Recall Kaun sa case kaun sa hai?

Jab $r \to 0$ , $F$ jaata hai ::: infinity ki taraf ( $1/ r^{2}$ diverge karta hai). Jab $r \to \infty$ , $F$ jaata hai ::: zero ki taraf, lekin dheere, $1/ r^{2}$ ki tarah. Agar koi bhi charge zero hai, $F$ hai ::: exactly zero — partner nahi, force nahi.

Ek-picture summary

Sab kuch ek saath: ek point charge apna influence ek sphere par spray karta hai ( $4 π r^{2}$ , jisse $1/ r^{2}$ milta hai); do charges ki maatraaein multiply hoti hain ( $q_{1} q_{2}$ ); ek constant $k$ scale set karta hai; product ka sign push ya pull choose karta hai.

$F_{21} = scale k product / sphere area (sign sets direction) \frac{q _{1} q _{2}}{r ^{2}} direction 1 \to 2 \overset{r}{^}_{21}$

Recall Feynman retelling — poori walkthrough plain words mein

Ek charged speck se shuru karo. Woh barabar sabhi directions mein bahar nikalti hai, jaise ek point se spray ki gayi paint. Do guna door jao aur woh paint ek ball ke upar failti hai jिसका skin char guna bada hai, toh woh char guna patli hai — yahi "one over r-squared" hai. Ab ek doosra speck lao: agar kisi bhi speck par charge double karo toh push double hoti hai, aur "dono mein se koi ek double hone par double hona" sirf multiplication hai, toh strength do charges multiplied ke saath jaati hai. Nature ek fixed dial add karta hai, jise $k$ kehte hain, us shape ko real newtons mein badalne ke liye. Aakhir mein, signs check karo: plus-aur-plus ya minus-aur-minus alag shove karte hain, plus-aur-minus snap together karte hain — product ka sign batata hai kaun sa. Unhe zero distance tak squeeze karo aur push infinity tak cheekhti hai; unhe infinitely door kheencho aur woh kuch nahi ho jaati lekin kabhi poori tarah nahi marti; kisi ek ko neutral banao aur force gayab ho jaati hai. Woh last fact hi poori wajah hai ki gravity kabhi jeet paati hai.