Superposition principle for forces — Hinglish notes

1.8.3 · Physics › Electromagnetism

Superposition principle KYA hai?

Key words hain vector sum (scalar nahi — directions matter karte hain) aur independent (charge $i$ ki force, charge $j$ ki presence se modify nahi hoti).

YEH SACH KYUN HAI? (first-principles reasoning)

YEH GALAT BHI HO SAKTA THA: agar forces "saturate" hoti (koi charge sirf itni hi total force feel kar sakta) ya agar ek charge doosre ko nonlinearly screen karta, toh superposition toot jaati. Yeh classical electrostatics mein extraordinary precision ke saath nahi toot-ti — isliye hum ise ek principle ki tarah adopt karte hain.

Net force KAISE derive karen (scratch se)

Hum kabhi seedha "plug in" nahi karte. Hum ise build karte hain.

Step 1 — EK pair ke liye Coulomb's law. $q_{0}$ par force jo position $r_{0}$ par hai, $q_{i}$ se jo position $r_{i}$ par hai: $F_{0 i} = \frac{1}{4 π ε _{0}} \frac{q _{0} q _{i}}{∣ r _{0} - r _{i} ∣ ^{2}} \overset{r}{^}_{0 i}, \overset{r}{^}_{0 i} = \frac{r _{0} - r _{i}}{∣ r _{0} - r _{i} ∣}$ Yeh step kyun? Unit vector $\overset{r}{^}_{0 i}$ source $i$ se target $0$ ki taraf point karta hai, isliye repulsion (same signs) automatically $q_{0}$ ko door push kar deta hai. Direction geometry se handle hoti hai, haath se nahi.

Step 2 — Superposition apply karo. Contributions add karo: $F_{net} = \frac{q _{0}}{4 π ε _{0}} i = 1 \sum n \frac{q _{i} ( r _{0} - r _{i} )}{∣ r _{0} - r _{i} ∣ ^{3}}$ Cube kyun? Humne $\overset{r}{^}_{0 i} = (r_{0} - r_{i}) /∣ r_{0} - r_{i} ∣$ likha, jisse distance ka ek extra power denominator mein aata hai, $r^{2}$ ko $r^{3}$ mein badal deta hai jabki numerator direction carry karta hai.

Step 3 — Components mein resolve karo. Kyunki yeh vector sum hai: $F_{x} = \sum_{i} F_{0 i} cos θ_{i}, F_{y} = \sum_{i} F_{0 i} sin θ_{i}$ $∣ F_{net} ∣ = F_{x}^{2} + F_{y}^{2}, tan ϕ = \frac{F _{y}}{F _{x}}$ Components kyun? Tum alag-alag directions mein point kar rahi forces ke magnitudes add nahi kar sakte. Components se tum safely har axis ke saath add kar sakte ho, phir recombine kar sakte ho.

Figure — Superposition principle for forces

Worked Example 1 — Do charges, dono x-axis par

Charges $q_{1} = + 3 μ C$ at $x = 0$ aur $q_{2} = - 2 μ C$ at $x = 4 m$ . $q_{0} = + 1 μ C$ at $x = 2 m$ par force nikalo. Maano $k = \frac{1}{4 π ε _{0}} = 9 \times 1 0^{9}$ .

$q_{1}$ se: distance $= 2$ m, same signs ⇒ repulsion ⇒ $q_{0}$ ko +x mein push karta hai. $F_{01} = \frac{k ( 3 \times 1 0 ^{- 6} ) ( 1 \times 1 0 ^{- 6} )}{2 ^{2}} = 6.75 \times 1 0^{- 3} N (+x)$ +x kyun? $q_{1}$ baayein hai aur repel karta hai, isliye $q_{0}$ ko daayein dhakelta hai.
$q_{2}$ se: distance $= 2$ m, opposite signs ⇒ attraction ⇒ $q_{0}$ ko $q_{2}$ ki taraf kheenchta hai (daayein) ⇒ +x. $F_{02} = \frac{k ( 2 \times 1 0 ^{- 6} ) ( 1 \times 1 0 ^{- 6} )}{2 ^{2}} = 4.5 \times 1 0^{- 3} N (+x)$ Phir +x kyun? $q_{2}$ daayein hai aur attract karta hai, $q_{0}$ ko daayein kheenchta hai.
Net: dono same direction mein ⇒ add karo: $F = 11.25 \times 1 0^{- 3} N = 11.25 mN$ along +x.

Worked Example 2 — Right-angle (2D vector sum)

$q_{1} = + 1 μ C$ at origin, $q_{2} = + 1 μ C$ at $(0, 3)$ m. $q_{0} = + 1 μ C$ at $(4, 0)$ m par force nikalo.

$q_{1}$ se: $r = 4$ m, +x along repulsion. $F_{01} = \frac{k ( 1 ) ( 1 ) \times 1 0 ^{- 12}}{16} = 5.625 \times 1 0^{- 4} N \Rightarrow (5.625 \times 1 0^{- 4}, 0)$
$q_{2}$ se: separation vector $(4, 0) - (0, 3) = (4, - 3)$ , magnitude $5$ m. $F_{02} = \frac{k \times 1 0 ^{- 12}}{25} = 3.6 \times 1 0^{- 4} N$ Direction = unit vector $(4, - 3) /5 = (0.8, - 0.6)$ . Yeh unit vector kyun? Yeh source $q_{2}$ se target $q_{0}$ ki taraf point karta hai — repulsion iske saath baahir dhakelta hai. $F_{02} = 3.6 \times 1 0^{- 4} (0.8, - 0.6) = (2.88 \times 1 0^{- 4}, - 2.16 \times 1 0^{- 4})$
Components sum karo: $F_{x} = (5.625 + 2.88) \times 1 0^{- 4} = 8.505 \times 1 0^{- 4}, F_{y} = - 2.16 \times 1 0^{- 4}$ $∣ F ∣ = 8.50 5^{2} + 2.1 6^{2} \times 1 0^{- 4} = 8.78 \times 1 0^{- 4} N, ϕ = arctan \frac{- 2.16}{8.505} = - 14. 3^{\circ}$

Arctan kyun? +x axis ke neeche ka angle, $q_{2}$ ke downward pull-component ki wajah se hone wala tilt record karta hai.

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum bheed mein khade ho aur teen dost tumhe ropes se push ya pull kar rahe hain. Friend A ko parwah nahi ki B aur C exist karte hain — woh tumhe exactly utni strength se kheechtey hain jitni woh akele use karte. Yeh find karne ke liye ki tum actually kahaan move karoge, tum saare pulls ko arrows ki tarah add karte ho: north pull aur east pull milkar tumhe northeast bhejtey hain. Yeh "arrows add karo" wala rule hi superposition principle hai.

Active Recall

Forces ke liye superposition principle kya kehta hai?

Kisi charge par net force, har doosre charge se aane wali individual Coulomb forces ka vector sum hoti hai, jisme har ek ko aise calculate kiya jaata hai jaise wahi akela source ho.

Superposition scalar operation hai ya vector?

Vector — directions ko magnitudes se nahi, components ke zariye add karna hota hai.

Kya teesra charge do charges ke beech ki pairwise force ko alter karta hai?

Nahi. Har pair force independent hoti hai; teesra charge sirf apna alag force term add karta hai.

Vector Coulomb form mein denominator mein

r^{3}

kyun hota hai?

\overset{r}{^} = (r_{0} - r_{i}) /∣ r_{0} - r_{i} ∣

likhne se distance ka ek extra power aata hai,

r^{2}

r^{3}

mein badal deta hai jabki numerator direction carry karta hai.

Force ON

q_{0}

ke liye

\overset{r}{^}_{0 i}

kis taraf point karta hai?

Source charge

q_{i}

se target

q_{0}

ki taraf.

Force magnitudes ko simply kab add kar sakte ho?

Sirf jab saari forces collinear hon (ek hi line par).

Kya superposition Coulomb's law se akele derive hoti hai?

Nahi — source charges mein linearity ek alag experimental fact hai jise Coulomb's law satisfy karta hai.

Connections

Coulomb's Law — woh single-pair force jise superposition add karta hai.
Electric Field — fields bhi isi superposition ko follow karte hain, $E = \sum E_{i}$ .
Vector Addition and Resolution — component summation ka math engine.
Continuous Charge Distributions — superposition ek integral $\int d F$ ban jaata hai.
Electric Potential — scalar superposition (zyaada simple, bas numbers add karo).
Principle of Linear Superposition (Waves) — same idea, alag physics.