1.8.10
Ek equipotential surface woh sheet hoti hai jis par har point ka same electric potential V E YEH INEVITABLE KYUN LAGTA HAI? Agar E along hota, toh woh component kisi charge ko surface ke andar push karta — kaam karta — aur ek charge ko do same-potential points ke beech move karne mein zero kaam lagta hai. Contradiction hai. Toh E
Definition Equipotential surfaceEk surface (points ka locus) jis par electric potential ka ek hi constant value hota hai:
V ( x , y , z ) = constant . along move karta hai toh electric force koi bhi kaam nahi karta.
Definition "Iske along kaam kyun nahi hota"Charge q A B W A B = q ( V A − V B ) . A , B V A = V B W A B === 0 == kisi bhi path ke liye.
Hum field aur potential ke beech sabse fundamental relation se shuru karte hain.
Step 1 — Work–potential link.
Ek chhoti displacement d l d V = − E ⋅ d l . Yeh step kyun? Yeh potential difference ki definition hai: V along move karte ho, isliye minus sign "potential E
Step 2 — d l andar restrict karo.
Ek equipotential par, V d V = 0 0 = − E ⋅ d l (tangent) . Yeh step kyun? Hum deliberately woh displacements choose karte hain jo surface par rehti hain, kyunki wahi moves hain jo V
Step 3 — Dot product ko interpret karo.
E ⋅ d l (tangent) = 0 har tangent d l ke liye. perpendicular hoon. Kyunki d l kisi bhi tangent direction mein point kar sakta hai, E sabhi ke perpendicular hona chahiye — yaani surface ke hi perpendicular.
E ⊥ equipotential surface Yeh step kyun? "Har in-surface direction ke perpendicular" — yahi toh "surface ka normal" ki definition hai.
Intuition Magnitude bonus: spacing strength batata haiDo equipotentials ke beech ek tiny perpendicular distance d n d V E = − d n d V . paas paas packed hain, E bada hota hai (steep potential hill); jahan woh door door hain, E
Worked example 1. Point charge — concentric spheresOrigin par + Q V = 4 π ε 0 1 r Q V = ⇒ r = Q spheres hain.
Field radial hai: E = 4 π ε 0 1 r 2 Q r ^ Perpendicular kyun? r ^ normal hai — toh E 9 0 ∘ Uneven spacing kyun? Equal Δ V r door hote jaate hain (kyunki V ∼ 1/ r E ∼ 1/ r 2
Worked example 2. Uniform field (parallel plates)E = E x ^ V = − E x + C V = ⇒ x = flat planes hain x ^ ⊥ Δ V equally spaced planes milti hain (kyunki V x E Yeh kyun matter karta hai: yeh explain karta hai ki charged parallel-plate capacitor ke andar evenly-spaced equipotential sheets kyun hoti hain.
Worked example 3. Electrostatics mein conductor ki surfaceConductor ke andar E = 0 V poora conductor ek equipotential hai, uski surface bhi.
Isliye bahar, E perpendicular hona chahiye. Agar nahi hota, toh tangential component free electrons ko push karta jab tak woh rearrange hokar use cancel na kar dete.
Yeh killer application kyun hai: isliye field lines metal surfaces ko exactly 9 0 ∘
Common mistake "Equipotentials aur field lines parallel hain."Kyun sahi lagta hai: dono diagrams mein "lines" dikhti hain, aur field lines clearly kaheen jaati hain , toh aap imagine karte ho ki equipotential saath-saath chal rahi hai.
Fix: woh perpendicular hain, parallel nahi. Field lines equipotentials ke across point karti hain (V field lines neeche jaati hain, equipotentials wrap karti hain.
Common mistake "Kaam surface ke along path par depend karta hai."Kyun sahi lagta hai: roz ki zindagi mein lambe paths par zyada kaam hota hai.
Fix: electric force conservative hai; W = q ( V A − V B ) V A = V B W = 0
Common mistake "Do alag equipotential surfaces touch/cross kar sakti hain."Kyun sahi lagta hai: contour lines saddle par milti hui lag sakti hain.
Fix: crossing point par ek saath do values of V V kabhi intersect nahi karti .
Common mistake "Equally spaced equipotentials ka matlab hamesha equal field hota hai."Kyun sahi lagta hai: map ka intuition.
Fix: sirf tab sach hai jab unhe equal Δ V par draw karo. Equal Δ V E Δ V E
Recall Feynman: ek 12-saal ke bachche ko explain karoEk pahaad imagine karo. Zameen ki height voltage jaisi hai. Woh curved lines jo "sab points same height par" trace karti hain, woh hiking map par wali contour lines hain — woh equipotentials hain. Ek ball hamesha seedha sabse steep raaste se neeche ludkti hai , jo exactly un same-height lines ke right angles par hoti hai. Woh steepest-downhill arrow electric field hai. Toh field aur equal-height lines hamesha ek +
"E cuts V at 90, downhill all the time."
( E lectric field equiV potentials ko right angles par cuts karta hai, lower potential ki taraf point karta hai. )
Equipotential surface kya hoti hai? Woh surface jis par electric potential V
Equipotential ke along charge move karne mein kaam kyun nahi hota? Kyunki W = q ( V A − V B ) V A = V B W = 0
E 9 0 ∘
Derive karo kyun E ⊥ Tangent step
d l ke liye,
d V = − E ⋅ d l = 0 ; nonzero vectors ka dot product sabhi tangent directions ke liye zero ⇒
E surface ka normal hai.
Field aur potential ka relation (vector form)? E = − ∇ V ; gradient constant-
V surfaces ke perpendicular hota hai aur higher
V ki taraf point karta hai, toh
E lower
V ki taraf point karta hai.
Equipotential spacing se field ka magnitude? E = − d n d V Δ V
Point charge ke liye equipotentials ki shape? Charge par centred concentric spheres.
Uniform field mein equipotentials ki shape? E ke perpendicular flat planes, equal
Δ V ke liye equally spaced.
Conductor ki surface equipotential kyun hoti hai? Andar
E = 0 toh
V poore andar constant hai; isliye surface bhi equipotential hai aur
E bahar uske perpendicular hota hai.
Kya alag V Nahi — crossing point par ek saath do potential values honge, jo impossible hai.
E Higher potential se lower potential ki taraf (V
along move karne mein koi kaam nahi
tangent tak restrict karo
gradient constant V ka normal
spacing strength deta hai
Equipotential surface V constant
W_AB = q times V_A minus V_B = 0
Koi in-surface E component nahi
E surface ke perpendicular
