Electric field — definition, field lines, superposition

1.8.4 · Physics › Electromagnetism

1. Definition

Limit $q_{0} \to 0$ kyun? Ek real test charge source charges ko push karta aur unhe move kar deta, wahi field badal deta jo hum measure karna chahte hain. Hum test charge ko itna vanishingly small imagine karte hain ki woh bina disturb kiye probe kare.

Point charge ka field derive karna — scratch se

WHAT hum jaante hain (Coulomb's law): point charge $q$ se $r$ distance par charge $q_{0}$ par force hai $F = \frac{1}{4 π ε _{0}} \frac{q q _{0}}{r ^{2}} \overset{r}{^} .$

HOW hum field nikalte hain: definition ke according test charge se divide karo. $E = \frac{F}{q _{0}} = \frac{1}{4 π ε _{0}} \frac{q q _{0}}{r ^{2} q _{0}} \overset{r}{^}$

Us point par rakhe kisi bhi charge $Q$ par force phir simply $F = Q E$ hoti hai. (Positive $Q$ ko $E$ ke direction mein force lagti hai; negative $Q$ ko, opposite direction mein.)

2. Field lines (Dual coding)

Figure — Electric field — definition, field lines, superposition

Density ka matlab strength kyun hai? Ek point charge ke liye, saari lines $4 π r^{2}$ area wale ek sphere par spread ho jaati hain. Jaise $r$ badhta hai, uthni hi lines ek bade sphere par spread hoti hain, toh density $\propto 1/ r^{2}$ — bilkul waise hi jaise $E$ fall off karta hai. Lines ki geometry automatically inverse-square law encode karti hai.

3. Superposition

Yeh allowed kyun hai? Coulomb's law charge mein linear hai: source double karo, force double ho jaati hai, aur forces khud vectors ki tarah add hoti hain (Newton). Linearity → fields add hoti hain. (Yeh sirf classical EM se bahar extreme regimes mein fail karta hai.)

4. Worked examples

Worked example (b) X-axis par

$\pm a$ par do equal charges $+ q$ — y-axis par height $y$ par field Step 1 — har charge se magnitude. Distance $r = a^{2} + y^{2}$ , toh $E_{1} = \frac{k q}{a ^{2} + y ^{2}}$ . Yeh $r$ kyun? Charge se point tak Pythagoras se.

Step 2 — components. Symmetry se dono horizontal $E_{x}$ cancel hote hain; vertical parts add hote hain. Vertical fraction hai $cos θ = \frac{y}{a ^{2} + y ^{2}}$ . Cancel kyun hote hain? Y-axis ke baare mein mirror symmetry $E_{x}$ ko flip karta hai par $E_{y}$ ko rakhta hai.

Step 3 — total. $E_{y} = 2 E_{1} cos θ = \frac{2 k q}{a ^{2} + y ^{2}} \cdot \frac{y}{a ^{2} + y ^{2}} = \frac{2 k q y}{( a ^{2} + y ^{2} ) ^{3/2}}$ $3/2$ power kyun? Coulomb se $r^{2}$ ka ek power, $cos θ$ projection se half power.

Limit check $y ≫ a$ : $E_{y} \to \frac{2 k q}{y ^{2}}$ — dur se single charge $2 q$ jaisa lagta hai. ✓ (Forecast-then-verify: humne predict kiya tha "dur se sirf total charge," aur wahi hai.)

5. Common mistakes

Recall Feynman: 12-saal ke bachche ko explain karo

Sochho har charged ball ek tiny heater ki tarah hai jo apne aas-paas ke room ko warm karti hai — tum "warmth" (force) feel karte ho bina heater ko touch kiye. Electric field ek map hai ki warmth kitni strong hai aur har jagah kaise push karti hai. Agar room mein do heaters rakh do, toh kisi bhi jagah warmth dono warmths ko ek saath add karne jaisi hai (arrows ki tarah). Paper par bane arrows field lines hain: crowded arrows = strong push.

Flashcards

Electric field

E

ki definition kya hai?

Force per unit positive test charge,

E = lim_{q_{0} \to 0} F / q_{0}

, units N/C.

Definition mein

q_{0} \to 0

ka limit kyun lete hain?

Taaki test charge source charges ko disturb (move) na kare aur measure ki ja rahi field na badle.

Point charge ka field?

E = \frac{1}{4 π ε _{0}} \frac{q}{r ^{2}} \overset{r}{^}

, jahan

\overset{r}{^}

source se field point ki taraf ho.

Field lines cross kyun nahi kar sakti?

Crossing par

E

ke do directions ho jaate, par field har point par exactly ek value rakhti hai.

Field-line density kya represent karti hai?

E

ki magnitude (strength); denser = stronger.

Superposition principle batao.

E_{n e t} = \sum_{i} E_{i}

— fields vectors ki tarah add hoti hain, doosre charges se independent.

Superposition valid kyun hai?

Coulomb's law charge mein linear hai aur forces vectors ki tarah add hoti hain.

\pm a

par do equal

+ q

ka, height

y

par on-axis field?

E_{y} = \frac{2 k q y}{( a ^{2} + y ^{2} ) ^{3/2}}

, horizontal parts cancel ho jaate hain.

Dipole ka far field

1/ r^{3}

se kyun fall karta hai?

+

aur

-

almost cancel ho jaate hain; sirf unki chhoti si separation bachti hai, falloff steep ho jaata hai.

Point-charge field formula mein

\overset{r}{^}

ki direction?

Source charge se field point ki taraf.

Connections

Coulomb's Law — woh force law jisse $E$ bana hai.
Gauss's Law — "line count" ko flux $Φ = \oint E \cdot d A$ ke through formalize karta hai.
Electric Potential — $E = - \nabla V$ ; field lines equipotentials ke perpendicular hoti hain.
Electric Dipole — example (c) iska starting point hai.
Continuous Charge Distributions — superposition ek integral ban jaata hai.
Superposition Principle — waves aur circuits mein bhi yahi linearity idea.