Electric potential — definition V = −∫E·dl

1.8.8 · Physics › Electromagnetism

KYA hai electric potential?

"Per unit charge" quantity kyun define karte hain? Kyunki kisi charge par force $q_{0}$ ke saath scale karti hai, isliye work bhi $q_{0}$ ke saath scale karti hai. Use divide karne par sirf field/location ka property milta hai — kuch aisa jo ek baar map karo aur kisi bhi charge ke liye reuse karo.

$V = - \int E \cdot d l$ KAISE milta hai — poora derivation

Formula First principles se Derivation

Step 1 — Test charge par Force. Field $E$ mein charge $q_{0}$ par $F_{elec} = q_{0} E$ lagti hai. Yeh step kyun? Yahi $E$ ki definition hai (force per unit charge).

Step 2 — Ise quasi-statically move karo. Ise constant KE par chalane ke liye, external agent $F_{ext} = - F_{elec} = - q_{0} E$ lagate hain. Kyun? Newton ka 2nd law: zero net force ⇒ kinetic energy mein koi change nahi, isliye saara kaam potential energy mein jaata hai.

Step 3 — Agent dwara kiya gaya Work ek path par $a$ se $b$ tak: $W_{ext} = \int_{a}^{b} F_{ext} \cdot d l = - q_{0} \int_{a}^{b} E \cdot d l$ Dot product kyun? Sirf displacement ke saath force ka component kaam karta hai.

Step 4 — $q_{0}$ se divide karo (potential difference work per charge hai): $V_{b} - V_{a} = - \int_{a}^{b} E \cdot d l$

Step 5 — Reference choose karo. $V = 0$ infinity par set karo (localized charges ke liye): $V (P) = - \int_{\infty}^{P} E \cdot d l$

Ulta relation: field potential ka gradient hai

Kyunki $d V = - E \cdot d l$ , toh $E \cdot d l = E_{x} d x + E_{y} d y + E_{z} d z$ likho. $d V = \frac{\partial V}{\partial x} d x + \dots$ se match karne par:

Figure — Electric potential — definition V = −∫E·dl

Worked examples

Forecast-then-Verify

Recall Compute karne se pehle Forecast karo

Q: Ek $+ 2 μ$ C charge $V = 50$ V wale point se $V = 20$ V wale point tak move karta hai. Kya field positive ya negative work karti hai, aur kitna? Pehle Forecast karo, phir kholो: Field dwara work $= q (V_{a} - V_{b}) = (2 \times 1 0^{- 6}) (50 - 20) = + 6 \times 1 0^{- 5}$ J. Positive — charge potential mein "neechey" gaya, bilkul ek ball ke neeche ludhakne ki tarah.

Common mistakes (Steel-man + fix)

Flashcards

Kisi point par electric potential kya hai?

Unit positive charge ko reference (∞) se us point tak laane mein kiya gaya work per unit charge, bina KE change kiye. Units: volts (J/C).

V aur E ke beech defining integral relation bolo.

V_{b} - V_{a} = - \int_{a}^{b} E \cdot d l

V = - \int E \cdot d l

mein minus sign kyun hai?

Kyunki integral field dwara kiya gaya work hai; potential us direction mein kam hota hai jahan E point karta hai (field "downhill" point karta hai).

Electric potential scalar hai ya vector?

Scalar — sirf magnitude hai; direction ki info

E = - \nabla V

mein hai.

Point charge ke liye V(r) derive karo.

V = - \int_{\infty}^{r} \frac{k q}{r ^{'2}} d r^{'} = k q / r

V se E kaise nikaalte hain?

E = - \nabla V

(scalar potential ka negative gradient).

Uniform field ke liye E, ΔV aur plate separation d ka relation bolo.

E = Δ V / d

, E high se low potential ki taraf point karta hai.

Field dwara charge q ko a se b tak move karne mein kiya gaya work?

W_{f i e l d} = q (V_{a} - V_{b})

V=0 at infinity kab fail karta hai?

Infinite charge distributions (line/plane) ke liye — integral diverge karta hai; finite reference use karo.

Recall Feynman: 12-saal ke bachche ko samjhao

Ek skateboard ramp socho. Potential bas ramp par kitna upar ho — zyada upar matlab zyada "stored go". Electric field ramp kitni steep hai hai, aur yeh hamesha neechey point karta hai. Agar tum kahi bhi khade ho, unchaai batati hai ki tum kitni energy release kar sakte ho. Formula $V = - \int E \cdot d l$ literally kehta hai: "unchaai find karne ke liye, chalte waqt steepness add karo, aur minus isliye ki jis taraf field push karta hai us taraf chalna matlab neeche jaana hai."

Connections

Electric field E — definition and Coulomb's law
Potential energy of a charge configuration
Equipotential surfaces and why E ⟂ them
Gradient operator ∇ and directional derivatives
Work–energy theorem in electrostatics
Capacitance and parallel-plate fields (E = ΔV/d)