Potential of point charge, potential from field and vice versa: Foundations

1.8.9 · D1 · Physics › Electromagnetism › Potential of point charge, potential from field and vice ver

Yeh page ek toolbox check hai. Parent note Potential of point charge, potential from field and vice versa ko follow karne se pehle, usmein use hone wala har single symbol obvious lagna chahiye. Hum unhe order mein banate hain — har ek sirf pehle wale symbols use karta hai.

1. Ek charge $q$ — har cheez ka source

Picture: ek dot jisme $+$ ya $-$ label hai jo space mein ek point pe baitha hai. Parent page pe sab kuch aise ek dot se bahar ki taraf radiate karta hai.

Topic ko yeh kyun chahiye: poori story yeh hai ki "ek charge $q$ apne aas-paas har jagah ek field aur ek potential create karta hai." $q$ ke bina kuch bhi create karne wala nahi hai. Iska sign answer $V = k q / r$ tak survive karta hai — ek negative charge negative potential deta hai.

Push ki size Coulomb's Law se set hoti hai, jis par hum aage lean karte hain.

2. Distance $r$ aur unit vector $\overset{r}{^}$ — kahan aur kis direction mein

Figure — Potential of point charge, potential from field and vice versa

Picture (upar wali figure): charge centre pe baitha hai. Koi bhi point $P$ chuno. Charge se $P$ tak seedhi line kheecho — uski length $r$ hai, aur chhota arrow $\overset{r}{^}$ us line par bahar ki taraf point karte hue baitha hai.

Dono kyun chahiye:

$r$ poochhta hai "kitna door?" — aur field aage jaane par weak hoti hai, isliye distance us weakening ka input hai.
$\overset{r}{^}$ poochhta hai "field kis direction mein point karti hai?" — ek positive charge ke liye field arrows seedhe bahar point karte hain, bilkul $\overset{r}{^}$ ke along.

Ek arrow ko "kitna lamba" ( $r$ ) aur "kis direction mein" ( $\overset{r}{^}$ ) mein split karna woh trick hai jo hume agle section mein field ko cleanly likhne deti hai.

3. Electric field $E$ — force per charge

Picture: charge ke aas-paas space ko bhar dene wale arrows ka poora field — charge ke paas lamba, door chhota, sab bahar ki taraf point karte hue ( $+ q$ ke liye) ya andar ki taraf ( $- q$ ke liye).

Ise teen factors multiply hote hue padho:

$\frac{q}{r ^{2}}$ — ek number jo batata hai kitna strong (charge ke saath badhta hai, distance-squared ke saath ghatta hai),
$\overset{r}{^}$ — direction (seedhe bahar),
$\frac{1}{4 π ε _{0}}$ — ek constant jo sirf units fix karta hai (agla section).

Topic ko yeh kyun chahiye: $E$ poori story ka aadha hissa hai. Parent page ka poora mission is arrow-field aur height-landscape $V$ ke beech convert karna hai. Yahi exact formula $V$ ki derivation mein integrate hota hai. Apni derivation ke liye Electric Field of Point Charge dekho.

4. Constants $ε_{0}$ aur $k$ — unit-fixers

Picture: $k$ ko ek fixed exchange rate socho. Charge aur distance answer ki shape decide karte hain; $k$ us shape ko real volts aur newtons mein convert karta hai.

Topic ko yeh kyun chahiye: $k$ ke bina numbers galat units mein aate hain. Parent page $V = k q / r$ likhta hai precisely isliye kyunki $\frac{1}{4 π ε _{0}}$ ko har jagah carry karna clumsy hai — $k$ wohi cheez hai, packaged form mein.

5. Test charge $q_{0}$ aur work $W$ — potential measure karna

Picture (upar wali figure): lal arrow field ka $q_{0}$ par push hai. Violet arrow aap hain jo $q_{0}$ ko door se andar ki taraf drag kar rahe ho. Jab aap push ke against drag karte ho, aap positive work karte ho — woh stored effort hi potential ban jata hai.

Topic ko yeh kyun chahiye: potential define hota hai work-per-charge ke roop mein: $V = W_{ext} / q_{0}$ . Parent page ki definition line ko samjhe bina "work by an external agent" ka matlab jaanne ke bina nahi samajh sakte. Yahi idea Potential Energy of Charge System mein kaam aata hai, jahan $U = q V$ .

6. Dot product $E \cdot d l$ — kitni field path ke along hai

Picture (upar wali figure): teen cases side by side. Same direction → full product. Angle par → sirf shadow (projection) count hoti hai. Perpendicular → kuch nahi, zero.

Topic ko yeh kyun chahiye: work force hai direction of motion ke along. Jab hum $q_{0}$ ko ek tiny step $d l$ (movement ka ek tiny arrow) se drag karte hain, sirf $E$ ka woh part jo $d l$ ke along point karta hai koi work karta hai. Dot product $E \cdot d l$ exactly woh part extract karta hai. Isliye parent ka radial-path trick kaam karta hai: $\overset{r}{^}$ ke along seedhe move karo aur $E \cdot d l = E d r$ — perfectly aligned, full product, koi wasted angle nahi.

7. Line integral $\int E \cdot d l$ — path ke along add karna

Picture: section 5 ka inward drag, lekin ab countless tiny steps mein slice kiya gaya, har ek thoda sa work contribute karta hai, sab sum hote hain.

Topic ko yeh kyun chahiye: potential ki definition hai ek line integral: $V (P) = - \int_{\infty}^{P} E \cdot d l$ "Infinity se $P$ tak field-slivers ko add karo, sign flip karo." Section 8 explain karta hai hum ek single aisa number likhne ki permission kyun rakhte hain.

8. Conservative field aur gradient — do-taraf ka bridge

Picture: ek hilly landscape. Base camp se peak tak jaane ke liye, koi bhi trail lo — height gained same hogi. Wapas start tak loop karo → zero net climb.

Topic ko yeh kyun chahiye: yeh poore potential idea ka permission slip hai. Sirf isliye kyunki har path same answer deta hai, hum har point ko ek clean number $V (P)$ assign kar sakte hain. Agar loops cancel nahi karte, to "height" ambiguous hoti. Poori story Conservative Fields and Curl mein hai.

Picture (upar wali figure): ek hill ki contour lines (equal height ki curves). Gradient arrow unhe right angles par cross karta hai, higher ground ki taraf point karta hai; yeh wahan sabse lamba hota hai jahan contours crowd hote hain.

Topic ko yeh kyun chahiye: reverse bridge hai $E = - \nabla V$ . Field = minus the uphill arrow = downhill arrow. Minus sign isliye hai ki field high potential se low potential ki taraf point karta hai. Gradient Operator aur Equipotential Surfaces mein perpendicular-to-contours fact dekho.

Prerequisite map

Equipment checklist

Right side cover karo aur khud test karo. Agar koi bhi answer surprise kare, upar woh section dobara padho.

q

ka sign final potential mein kya decide karta hai?

V = k q / r

positive aata hai (

+ q

ke liye) ya negative (

- q

ke liye).

r

aur

\overset{r}{^}

alag-alag kya do pieces of information carry karte hain?

r

= kitna door (ek length);

\overset{r}{^}

= kis direction mein (sirf direction, length 1).

E

ek phrase mein kya hai, aur uski units?

Force per unit positive charge; units N/C = V/m.

k

kya hai aur uski approximate value?

k = 1/ (4 π ε_{0}) \approx 9 \times 1 0^{9} N m^{2} / C^{2}

, unit-fixing constant.

Jab aap ek charge ko field ke against drag karte ho, aapka kiya work positive hai ya negative?

Positive — aap field se fight karte ho, energy potential ke roop mein store hoti hai.

Dot product

E \cdot d l

kya deta hai jab

E ⊥ d l

Zero — perpendicular arrows koi shadow nahi dalte.

Line integral

\int E \cdot d l

physically kya add up karta hai?

Start se end tak ke saare tiny field-along-the-path slivers.

Hum har point ko ek single number

V (P)

kyun assign kar sakte hain?

Kyunki field conservative hai — har path same work deta hai, isliye height ambiguous nahi hai.

\nabla V

kis direction mein point karta hai, aur

E = - \nabla V

kis direction mein point karta hai?

\nabla V

uphill point karta hai (badhta

V

);

E

downhill point karta hai (girta

V

Connections

Coulomb's Law — $q$ ki push ki strength set karta hai.
Electric Field of Point Charge — woh $E$ jo hum integrate karte hain.
Potential Energy of Charge System — jahan work $W = q V$ jaata hai.
Equipotential Surfaces — $V$ landscape ki contour lines.
Conservative Fields and Curl — potential ka permission slip.
Gradient Operator — $E = - \nabla V$ ke peeche ki machinery.