1.8.4 · D1Electric field — definition, field lines, superposition
Ek charge apne aas-paas ki space ko quietly badal deta hai, har point par ek ready-made "push per unit of charge" store karta hai — wahi stored push electric field hai. Parent page par jo bhi baaki hai woh sirf us push ko ek arrow ke roop mein likhne ka toolkit hai (magnitude + direction) aur aisi kai arrows ko saath add karna hai.
Parent note padhne se pehle, aapke paas pehle se kuch symbols aur pictures hone chahiye. Yeh page unhe ek-ek karke bilkul zero se build karta hai, usi order mein jisme woh ek-doosre par depend karte hain. Agar koi symbol parent page par aata hai, toh usse yahan pehle unpack kiya gaya hai.
Definition Electric chargeq positive ( + ) negative ( − ) coulombs mein measure hota hai (symbol C Picture: ek chota dot jisme + −
Topic ko iska zaroorat kyun hai: charge hi sab kuch ka source hai. Koi charge nahi, toh koi field nahi.
Ek rule jo aapko space mein har point par ek value deta hai. Ek scalar field aapko ek number deta hai (jaise room mein temperature). Ek vector field aapko poora ek arrow deta hai (ek size aur ek direction).
Picture: room mein kahin bhi khade ho jaao; room aapko silently ek number (temperature) ya ek arrow (hawa) bata deta hai.
Neeche di gayi figure yahi contrast draw karti hai. Board ke left half mein, har grid point par ek akela pale-yellow number hai — yeh ek scalar field hai (imagine karo har spot par temperature padhna). Right half mein, har grid point par ek chota blue arrow hai jiske paas length bhi hai aur heading bhi — yeh ek vector field hai (imagine karo hawa ki strength aur direction feel karna). Dotted vertical chalk line dono duniyaon ko alag karti hai. Electric field right side par rehta hai: har point par ek arrow.
Topic ko iska zaroorat kyun hai: electric field ek vector field hai. "Us point par field" kehne ke liye pehle yeh manna hoga ki space har jagah par ek arrow carry kar sakti hai.
Poora topic arrows ka hai. Isliye hume ek arrow ki vocabulary chahiye.
Definition Vector aur arrow-hat vector ek arrow hai: iske paas ek length hoti hai (kitna strong) aur ek direction hoti hai (kis taraf). Hum ek letter ke upar ek chota arrow rakhte hain yeh kehne ke liye ki "yeh ek vector hai": E F r Picture: kaagaz par kheencha gaya ek arrow. Uski length = strength; jis taraf uski tip point karti hai = direction.
Definition Magnitude — same letter bina hat keArrow ki sirf length (ek plain positive number, koi direction nahi). Hum ise bina arrow ke likhte hain: E E F F Picture: arrow ko ruler se measure karo; woh number magnitude hai.
^ ^ ^ positive x point karta hai (right taraf), ^ positive y point karta hai (upar). Har arrow tab "itna right plus itna upar" hota hai: A = A x ^ + A y ^ Picture: do chote rulers, ek floor par aur ek wall par chipke; har arrow unse measure hota hai.
r ^ point karna hai. Iski length exactly 1 direction carry karta hai, kuch aur nahi . Hat ^ r ^ r ^ ( x , y ) r ^ = r x ^ + r y ^ , r = x 2 + y 2 , ( x / r ) 2 + ( y / r ) 2 = 1 Picture: ek stubby arrow of fixed length 1 E r ^ ×
Intuition Har arrow ko (magnitude)× kitna strong (jo q kis taraf (jo geometry se milta hai). E = E r ^
Topic ko iska zaroorat kyun hai: parent ka headline formula E = 4 π ε 0 1 r 2 q r ^ × r ^ r ^ ^ , ^ x y
Definition Source point aur field pointSource point woh hai jahan charge hai (woh field banata hai). Field point woh hai jahan aap pooch rahe ho "yahan field kya hai?".
Picture: ek dot charge hai (source); doosra dot woh hai jahan tumhara measuring probe hai (field point).
r Picture: charge se probe tak kheenchi gayi ek tight string; uski length r
Neeche ki figure board par yeh triangle banati hai. Origin par pink dot source charge + q a y r r ^
r ^ r ^ source charge se field point ki taraf point karta hai — string charge ko chhodne ke baad jis direction mein point karti hai. Is figure mein r ^ = r a ^ + r y ^ Picture: charge par khade ho jaao, probe ko dekho; r ^
r ^ r ^ ki taraf point karte hain kyunki attraction "khenchne" jaisi lagti hai.
Fix: r ^ bahar jaata hai. q r ^
Intuition 2-D distance se 3-D sphere takAbhi humne triangle board par flat draw ki (ek 2-D slice) sirf number r compute karne ke liye. Lekin space 3-D hai: charge se fixed distance r saare points milke ek poori sphere banate hain radius r r r = x 2 + y 2 + z 2 4 π r 2
Topic ko iska zaroorat kyun hai: har field formula r ^
π ≈ 3.14159 sphere ke upar baahar bhejta hai, aur spheres round hote hain.
Picture: ek circle; π = 4 π r 2
ε 0 ε 0 ≈ 8.85 × 1 0 − 12 kitna strongly respond karti hai — ise "vacuum ki stiffness" samjho. Iska value charges-aur-distances ko real newtons mein convert karta hai.
Picture: universe par ek dial jo saare electric effects ki overall loudness fix karta hai.
Topic ko iska zaroorat kyun hai: 4 π ε 0 1 har field aur force formula ke aage baithta hai. Yeh ek "sphere par phailta hai" bookkeeping factor hai, kuch zyada scary nahi.
Field ko "force per charge" define karne se pehle, hume actual force chahiye. Woh force Coulomb's Law hai.
Intuition Jo pieces aapke paas pehle se hain unhe padheinYahan har symbol woh hai jo humne abhi banaya: k q 1 q 2 1/ r 2 r ^
r 2 1/ r 2 r 2 r × r 1/ r 2 r − 2 r 1/ r 2 fast ghatta hai: distance double karo, field quarter ho jaata hai.
Picture: wahi mutthi bhar arrows (field lines) ek sphere cover karne par majboor. Sphere ka area 4 π r 2
Neeche ki figure "spreading" ko literal banati hai. Centre mein pink charge saari directions mein fixed number of white field lines chhodta hai. Do chalk circles — ek chota blue wala radius r 2 r shells hain jinhe yeh lines cross karti hain. Wahi lines dono cross karti hain, lekin outer shell ka area chaar guna hai (area r 2 hi inverse-square law hai.
1/ r 2 1/ r surface par phailti hai — aur, jaise humne §2 mein kaha, real 3-D space mein woh surface ek poori sphere hai jiska area r 2 r 1/ ( area ) = 1/ r 2
Topic ko iska zaroorat kyun hai: ek point charge ke field ki poori strength, aur field-line density ka matlab strength kyun hai, dono isi 1/ r 2
F newtons ( N ) Picture: ek arrow jo probe charge ko dhakelta hai.
q 0 bahut chota positive charge jo hum field feel karne ke liye drop karte hain. Subscript 0 Picture: field point par rakhа gaya ek whisper-small +
lim q 0 → 0 q 0 → 0 lim Picture: thermometer ko chhota se chhota karte jaao taaki woh jis room ko maap raha hai use warm karna band kar de.
Topic ko iska zaroorat kyun hai: yeh poore subject ki definition hai. Har doosra formula is idea ko specific charges ke liye work out karta hai.
θ θ positive x ^ maapa jaata hai. Isliye θ = 0 θ = 9 0 ∘ θ = 18 0 ∘ θ = 27 0 ∘ cos θ sin θ Picture: origin par laga ek protractor jisme 0 ∘ x
E x E y horizontal (x (E x vertical (y (E y E = E x ^ + E y ^ Picture: seedha neeche light maaro — floor par shadow E x E y
cos θ sin θ θ + x E E x = E cos θ E y = E sin θ cos θ / sin θ / θ E x , E y signs har quadrant mein correctly aate hain.
Picture: arrow right triangle ki slanted side hai; cos θ
Topic ko iska zaroorat kyun hai: parent ke worked examples (b) aur (c) mein, fraction cos θ = a 2 + y 2 y
∑ i ∑ i ( stuff i ) i stuff 1 + stuff 2 + … i Picture: ek bucket jisme aap har charge ka arrow daalo, phir unhe tip-to-tail add karo.
Definition Integral symbol∫ d q ∫ ∑ Continuous Charge Distributions mein hai.)
Picture: arrows ka discrete bucket, lekin dots ek smooth smear ban jaate hain aur sum ek smooth total.
Intuition Hume arrows simply add karne kipermission kyun hai?
Kyunki Coulomb's Law charge mein linear hai (charge double karo, field double) aur arrows tip-to-tail add hote hain (Newton). Woh permission Superposition Principle hai — parent page ka teesra pillar.
Yahan pieces kaise stack hote hain, upar se neeche padho: raw materials (charge, arrows, distance) upar hain; unhe combine karke Coulomb's force banta hai; us force ko probe charge se divide karne par field define hoti hai; aur fields ko components mein todhne par hum kai fields add kar sakte hain — yahi poora parent topic hai. (Har box ek plain-English idea hai; arrows ka matlab hai "banane ke liye zaroori hai".)
Unit vector r-hat direction
Inverse square one over r squared
Point charge field formula
k equals one over four pi eps0
Force per test charge defines E
Electric field lines and superposition
Right side cover karo aur dekho ki reveal karne se pehle har ek bata sakte ho ya nahi.
r ^ r ^ Yeh ek unit vector mark karta hai — ek pure direction of length exactly 1
r ^ Source charge se field point ki taraf (charge se bahar).
Field point ( x , y ) r ^ x , y r ^ = r x ^ + r y ^ jahan
r = x 2 + y 2 .
E E E poora arrow hai (size + direction);
E bas uski length (magnitude) hai.
Coulomb's Law poora state karo. F = k r 2 q 1 q 2 r ^ ,
r ^ q 1 se
q 2 ki taraf;
q 1 q 2 ka sign attract/repel set karta hai.
k k = 4 π ε 0 1 ≈ 8.99 × 1 0 9 N m 2 / C 2
Field strength 1/ r 2 Wahi field ek 3-D sphere par phailti hai jiska area 4 π r 2 1/ r 2
E ≡ F / q 0 Force per unit test charge — probe ka size strip out karo taaki space ki property mile.
q 0 → 0 Taaki tiny probe source charges ko hilaye nahi aur jo field measure kar raha hai use corrupt na kare.
θ Positive x
Agar angle θ E E x = E cos θ E y = E sin θ
∑ i E i Har charge ke field arrow ko tip-to-tail add karo (superposition sum).
∫ r 2 d q r ^ ∑ Ek continuous smear ke infinitely many infinitesimal charge pieces add karta hai.
Hum fields add karne ki permission kyun rakhte hain? Coulomb's law linear hai aur forces vectors ki tarah add hote hain — superposition principle.
Yeh bhi dekho: Coulomb's Law , Superposition Principle , Continuous Charge Distributions , Electric Dipole , aur parent topic note .
