Visual walkthrough — Superposition principle for forces

1.8.3 · D2 · Physics › Electromagnetism › Superposition principle for forces

Hum is target ko chase kar rahe hain (ghabrao mat — last step tak yeh obvious lagega): ${\vec{F}}_{\text{net}}=\frac{q_0}{4\pi {\epsilon }_0}{\sum }_{i=1}^n\frac{q_i({\vec{r}}_0-{\vec{r}}_i)}{|{\vec{r}}_0-{\vec{r}}_i{|}^3}$

Step 1 — Players ko draw karo: charges positions par rehte hain

KYA. Kisi bhi force se pehle, hume ek tarika chahiye yeh kehne ka ki kahan har charge baitha hai. Hum ek grid ($x$–$y$ map) drop karte hain aur har charge ko origin se us dot tak ek arrow ke saath mark karte hain. Us arrow ko position vector kehte hain.

KYUN. Force ki ek direction hoti hai, aur direction tab hi samajh aati hai jab hum jaanein ki cheezein kahan hain. Akele numbers ("charge 2 metre door hai") direction kho dete hain; ek arrow distance aur direction dono rakhta hai.

PICTURE. Figure dekho. Target charge $q_0$ (blue dot) ka position arrow ${\vec{r}}_0$ hai. Ek source charge $q_i$ (orange dot) ka apna arrow ${\vec{r}}_i$ hai. Dono arrows origin $O$ se start hote hain.

Abhi kuch physical nahi — bas kahan ka bookkeeping. Aage hum ek charge ko doosre ko feel karaayenge.

Step 2 — Separation arrow ${\vec{r}}_0-{\vec{r}}_i$

KYA. Hum ek brand-new arrow banate hain jo source $q_i$ se seedha target $q_0$ ki taraf point karta hai. Hum ise position vectors subtract karke paate hain: ${\vec{r}}_0-{\vec{r}}_i$.

KYUN subtraction, aur kyun yeh order? Arrows subtract karna "tail se tip tak kaise pahunchoon" operation hai: ${\vec{r}}_0-{\vec{r}}_i$ woh arrow hai jo tum $q_i$ se start karke, $q_0$ par khatam karke chaloge. Hum yeh order isliye chahte hain kyunki jo force hum compute karne wale hain woh $q_0$ par act karti hai, toh woh $q_0$ ki taraf andar ki line ke saath point karni chahiye.

PICTURE. Figure mein, tip-to-tail: green arrow ${\vec{r}}_0-{\vec{r}}_i$ triangle ko close karta hai, orange source dot se blue target dot tak pahunchta hai. Iska length charges ke beech ki physical distance hai.

Step 3 — Pure direction nikalo: unit vector ${\hat{r}}_{0i}$

KYA. Hum separation arrow ko tab tak shrink karte hain jab tak uski length exactly $1$ na ho jaaye, same direction rakhte hue. Yeh kaam hum arrow ko uski apni length se divide karke karte hain. Result ek chhoti hat pehnta hai: ${\hat{r}}_{0i}$.

KYUN. Ek force ko ek strength aur ek direction chahiye, aur yeh dono alag-alag handle karna sabse clean hai. Unit vector "pure direction" hai — ek compass needle jiska apna koi length nahi. Baad mein hum ise strength (ek plain number) se multiply karte hain full force arrow rebuild karne ke liye.

PICTURE. Figure mein lambi green separation arrow dikhi hai aur, uske upar, length $1$ ka chhota red unit arrow same direction mein point karta hua dikhaya gaya hai. Same heading, standardised length.

Step 4 — Ek pair ke liye Coulomb's law (strength × direction)

KYA. Ab strength attach karo. Coulomb's Law do charges ke beech push ka size deta hai; hum us size ko pure-direction arrow ${\hat{r}}_{0i}$ se multiply karte hain taaki $q_0$ par sirf $q_i$ se full force arrow ${\vec{F}}_{0i}$ mile.

KYUN yahi exact strength? Experiment: pull/push har charge ke saath badhta hai (charge double karo, force double) aur distance ke square ke saath kamzor hota hai (twice as far → one-quarter force). Constant $\frac{1}{4\pi {\epsilon }_0}$ bas units fix karta hai (yaad karo ${\epsilon }_0$ vacuum permittivity hai jo upar define ki).

PICTURE. Figure mein full force arrow ${\vec{F}}_{0i}$ $q_0$ par baitha dikhaya gaya hai, red unit direction ke along point karta hua, strength ke proportional length ke saath. Do mini-panels repel (arrow bahar) aur attract (arrow andar) cases contrast karte hain.

Step 5 — Superposition principle: bahut saare arrows, ek crowd

KYA. Ek doosra source $q_j$ add karo. Yeh $q_0$ par apna force arrow ${\vec{F}}_{0j}$ produce karta hai, bilkul Step 4 ki tarah compute kiya, $q_i$ ko poori tarah ignore karke. Yeh "har source akele act karta hai jaise koi aur hai hi nahi" rule superposition principle hai.

KYUN yeh ek alag fact hai. Coulomb's law sirf do charges ka zikr karta hai — usme teesre ke liye koi jagah nahi. Toh do-charge forces kaise combine hoti hain yeh ek extra experimental law hai: nature linear nikli — forces bas add ho jaati hain, koi charge doosre ko screen ya kamzor nahi karta. Yeh aur bhi ho sakta tha; measurement kehti hai aisa nahi hai.

PICTURE. Teen source charges apne-apne colored arrows $q_0$ par fire karte hain. Khas baat yeh hai ki har arrow ki length aur heading sirf uske apne source se fix hoti hai — arrows ek doosre ko "notice" nahi karte.

Step 6 — Arrows sahi tarike se add karna: tip-to-tail

KYA. Sum actually karne ke liye, har force arrow ko tip-to-tail rakkho. Sabse pehle tail se sabse aakhri tip tak single arrow ${\vec{F}}_{\text{net}}$ hai. Yeh Vector Addition and Resolution in action hai.

KYUN bas lengths add nahi karte? Kyunki alag-alag directions mein arrows partly cancel karte hain. Right push aur upar push milke nahi banate (right + up) size ka push; woh ek diagonal push banate hain, aur uski length dono lengths add karne se kam hoti hai. Sirf same line par arrows ke numbers seedhe add ho sakte hain.

PICTURE. Figure Step 5 ke teen arrows ki tip-to-tail chain build karta hai aur mota resultant ${\vec{F}}_{\text{net}}$ polygon close karta hua dikhata hai.

Step 7 — Components mein resolve karo, phir recombine

KYA. Har force arrow ko $x$-axis par ek shadow aur $y$-axis par ek shadow do — uske components. Saare $x$-shadows add karke $F_x$ pao; saare $y$-shadows add karke $F_y$ pao. Phir un do totals se single arrow rebuild karo.

KYUN. Ek axis par shadows sab ek line par hain, toh woh plain numbers ki tarah add ho sakte hain. Yeh woh trick hai jo messy arrow-addition ko ordinary arithmetic mein badal deti hai jo do baar ki jaati hai.

PICTURE. Figure ek force arrow ke dashed shadows dono axes par girata hai, angle ${\theta }_i$ label karta hai, aur final ${\vec{F}}_{\text{net}}=(F_x,F_y)$ apne tilt angle $\varphi$ ke saath rebuild hota dikhata hai.

Step 8 — Degenerate cases (kabhi surprised mat hona)

KYA & KYUN. Ek achhi derivation apne edge cases survive karni chahiye. Check karo:

• Collinear charges (sab ek line par). Har arrow ka doosri axis par shadow zero hai, isliye $F_y=0$ aur tum do bas $x$ ke along signed numbers add karte ho. General recipe plain addition mein collapse ho jaati hai — exactly woh 1-D case.
• Ek source $q_0$ ke same spot par (${\vec{r}}_i={\vec{r}}_0$). Tab $|{\vec{r}}_0-{\vec{r}}_i|=0$ aur hum zero se divide karte hain — formula blow up ho jaata hai. Physically do point charges ek point par nahi baith sakte; yeh ek warning hai, koi value nahi.
• Ek zero charge ($q_i=0$). Uska term $0$ hai; kuch contribute nahi karta. Superposition quietly khali seats ignore karta hai.
• Exact cancellation. Do equal arrows opposite directions mein point karte hue zero arrow mein add ho jaate hain: ${\vec{F}}_{\text{net}}=\vec{0}$. Charge ko koi net force feel nahi hoti chahe har source hard push kar raha ho. (Uska angle $\varphi$ tab undefined hai — koi arrow point karne ke liye nahi.)

PICTURE. Chaar mini-panels: collinear (numbers add karo), coincident (explosion symbol), zero charge (ghost dot), aur perfect cancellation (arrows annihilate).

Ek-picture summary

Har step, compressed: positions → separation arrow → unit direction → one-pair force → saare sources ke liye repeat → tip-to-tail sum → components → net arrow.

$\boxed{{\vec{F}}_{\text{net}}=\frac{q_0}{4\pi {\epsilon }_0}\sum _{i=1}^n\frac{q_i({\vec{r}}_0-{\vec{r}}_i)}{|{\vec{r}}_0-{\vec{r}}_i{|}^3}}$

Active Recall

Connections

• Superposition principle for forces — woh parent jise yeh walkthrough visualise karta hai.
• Coulomb's Law — Step 4 mein build ki gayi one-pair force.
• Vector Addition and Resolution — Steps 6–7, arrow arithmetic.
• Electric Field — wahi superposition, forces ki jagah fields ke liye.
• Continuous Charge Distributions — sum ek integral ban jaata hai.
• Electric Potential — scalar superposition (bas numbers add karo).
• Principle of Linear Superposition (Waves) — same idea, alag physics.

Concept Map