Series and parallel capacitors — derivations: Question bank

1.8.12 · D5 · Physics › Electromagnetism › Series and parallel capacitors — derivations

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Parallel → har capacitor par same voltage $V$ ; charges add hote hain.
Series → har capacitor se same charge $Q$ guzarta hai; voltages add hote hain.

Yahan $V$ ka matlab hai potential difference (capacitor ke across "electrical pressure"), $Q$ ka matlab hai stored charge, aur dono [[Capacitance and the relation Q = CV| $Q = C V$ ]] se jude hain. Agar kisi line mein koi symbol samajh na aaye, to yahi relation usse define karti hai.

Shuru karne se pehle do wiring pictures dekho: woh dikhate hain ki har case mein kya physically shared hota hai (woh ek idea jis par neeche ke saare traps tike hain).

Figure — Series and parallel capacitors — derivations

Yeh page parent derivations note se juda hai — formulas wahan derive kiye gaye hain; yahan hum sirf understanding ko stress-test kar rahe hain.

True or false — justify

True or false: Series combination mein equivalent capacitance kisi ek member se bada ho sakta hai.

False. Reciprocals

\frac{1}{C _{1}} + \frac{1}{C _{2}} + \dots

add karne se

\frac{1}{C _{e q}}

kisi bhi single

\frac{1}{C _{i}}

se bada ho jaata hai, isliye

C_{e q}

sabse chhote member se bhi neeche chala jaata hai — kisi se bhi upar nahi.

True or false: Parallel mein, sabse bada capacitor sabse zyada charge store karta hai.

True. Sabka same

V

hota hai, aur

Q = C V

, isliye

Q \propto C

— zyada capacitance proportionally zyada charge leta hai.

True or false: Series mein, sabse bada capacitor sabse zyada charge store karta hai.

False. Series mein har capacitor same charge

Q

carry karta hai (beech ke plates isolated islands hain jo neutral rehte hain). Size voltage ko affect karta hai, charge ko nahi.

True or false: Parallel mein ek aur capacitor add karna total capacitance ko hamesha badhata hai.

True. Parallel mein

C_{e q} = \sum C_{i}

hota hai; har extra term positive hai, isliye sum sirf badh sakta hai (tumne charge hold karne ke liye aur plate area add kar diya).

True or false: Series mein ek aur capacitor add karna total capacitance ko hamesha ghataata hai.

True. Har extra positive

\frac{1}{C _{i}}

\frac{1}{C _{e q}}

ko badhata hai,

C_{e q}

ko neeche push karta hai (tumne ek aur gap stack kar diya, use "mota" aur fill karna mushkil bana diya).

True or false: Series mein do identical capacitors ek capacitor ki exactly aadhi capacitance dete hain.

True.

\frac{1}{C _{e q}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C}

, isliye

C_{e q} = C /2

— ek saaf, yaad rakhne laayak special case.

True or false: Series mein voltage hamesha do capacitors ke beech equally split hoti hai.

False. Yeh equally tabhi split hoti hai jab capacitances equal hon. Warna

V = Q / C \propto 1/ C

, isliye chhota capacitor zyada voltage leta hai.

True or false: Capacitors wahi series/parallel rules follow karte hain jo resistors follow karte hain.

False — yeh bilkul ulta hai. Resistor networks mein series resistances add hoti hain aur parallel wale reciprocals lete hain; capacitors mein roles swap ho jaate hain — parallel mein add karo, series mein reciprocals.

True or false: Equivalent capacitor wahi total charge store karta hai jo network battery se leta hai.

True — yahi equivalent ki definition hai,

C_{e q} = Q_{t o t a l} / V_{t o t a l}

. Woh same terminal voltage par same charge leta hai.

Spot the error

"Series mein battery

C_{1}

par

Q_{1}

charge push karti hai aur

C_{2}

par alag

Q_{2}

charge." — galti dhundho.

Middle conductor (

C_{1}

ki right plate jo

C_{2}

ki left plate se judi hai) ek isolated island hai. Charge conservation se woh neutral shuru hota hai aur usme koi charge enter ya leave nahi kar sakta, isliye ek plate par

+ Q

hone se agli plate par exactly

+ Q

force hoti hai — isliye

Q_{1} = Q_{2} = Q

, alag nahi.

"Series capacitors ke liye main bas inhe add karta hoon:

C_{e q} = C_{1} + C_{2}

." — galti dhundho.

Yeh parallel rule hai jo galti se resistors se transplant kiya gaya hai. Series capacitors mein reciprocals add hote hain:

\frac{1}{C _{e q}} = \frac{1}{C _{1}} + \frac{1}{C _{2}}

"Parallel mein har capacitor total voltage ka ek hissa leta hai jo milkar

V

banta hai." — galti dhundho.

Parallel mein har capacitor poora

V

feel karta hai (woh same do nodes / equipotential wires ke across hain). Charges milkar total banate hain, voltages nahi.

2 μ F

aur

3 μ F

ka series equivalent

\frac{2 + 3}{2 \cdot 3} = \frac{5}{6} μ F

hai." — galti dhundho.

Product-over-sum shortcut

\frac{C _{1} C _{2}}{C _{1} + C _{2}}

hai, yaani product upar hota hai. Student ne ise ulta kar diya. Sahi value

\frac{6}{5} = 1.2 μ F

hai.

"Kyunki series mein same charge flow karta hai, isliye har capacitor same energy store karta hai." — galti dhundho.

Energy formula ke hisaab se stored energy

\frac{Q ^{2}}{2 C}

hai;

Q

common hone par yeh

1/ C

ke saath scale karta hai, isliye chhota capacitor zyada energy store karta hai. Same charge ≠ same energy.

"Series chain mein ek capacitor mein dielectric daalna sirf usi capacitor ko affect karta hai, aur kuch nahi." — galti dhundho.

Dielectric us capacitor ka

C_{i}

badhata hai, jo uska

1/ C_{i}

term ghataata hai, isliye

C_{e q}

badhta hai, total charge

Q = C_{e q} V

badhta hai, aur har capacitor ki voltage

Q / C

shift ho jaati hai. Poori chain re-balance ho jaati hai.

Why questions

Parallel capacitors ke liye "same voltage" automatically kyun aata hai?

Har capacitor ki dono plates same do wires (nodes A aur B) se connect hain. Wire ek equipotential hoti hai, isliye har capacitor ke across potential difference forcibly same value ke barabar hota hai — battery voltage.

Parallel mein total charge branch charges ka sum kyun hota hai,

Q_{t o t a l} = \sum Q_{i}

Node A par battery se aane wali wire branches mein split hoti hai. Charge conservation se har second aane wala charge branches mein fan out hone wale charge ke barabar hona chahiye, isliye deliver kiya gaya total charge exactly branch charges ka sum hai:

Q_{t o t a l} = Q_{1} + Q_{2} + \dots

Series mein reciprocals kyun add hote hain, capacitances nahi?

Kyunki series mein shared quantity

Q

hai aur voltages add hote hain:

V = \sum Q / C_{i} = Q \sum 1/ C_{i}

Q

se divide karne par

1/ C_{e q} = \sum 1/ C_{i}

milta hai. Reciprocal isliye aata hai kyunki

C

V = Q / C

ke denominator mein hai.

Series mein chhota capacitor zyada voltage kyun leta hai?

Q

sabke liye common hone par,

V = Q / C

ka matlab hai

V \propto 1/ C

. Chhota

C

bada

V

banata hai — "tang bucket" ko same charge ke liye zyada pressure chahiye.

Series loop mein voltage drops ko add karke battery voltage kyun nikal sakte hain?

Kirchhoff's voltage law ki wajah se: potential single-valued hota hai, isliye loop mein ek baar ghoomne par drops applied EMF ke barabar hone chahiye,

V = V_{1} + V_{2} + \dots

Series equivalent hamesha sabse chhote member se chhota kyun hota hai?

Har extra reciprocal term sirf

1/ C_{e q}

ko badhata hai, isliye

1/ C_{e q}

har individual

1/ C_{i}

se zyada ho jaata hai, jiska matlab hai

C_{e q}

har individual

C_{i}

se neeche hai — chhote se bhi.

Series mein capacitors stack karna thicker dielectric gap banane jaisa kyun behave karta hai?

Capacitance plate separation badhne par ghatti hai. Series capacitors apne gaps ko end-to-end rakhte hain, isliye charge ko zyada total "insulating distance" cross karni padti hai, combined charge store karne ki ability ghatti hai.

Edge cases

Jab parallel group mein ek capacitor ka

C = 0

ho (ek ideal open gap) to equivalent kya hoga?

Woh sum

C_{e q} = \sum C_{i}

mein

0

contribute karta hai, isliye aisa hai jaise woh tha hi nahi — baaki sab sab kuch determine karte hain. Zero-capacitance branch simply koi charge store nahi karta.

Jab series chain mein ek capacitor ka

C \to 0

ho to equivalent kya hoga?

Uska term

1/ C \to \infty

sum mein dominate karta hai, isliye

1/ C_{e q} \to \infty

aur

C_{e q} \to 0

. Ek "fill hi nahi ho sakta" element poori chain ko nearly zero capacitance par choke kar deta hai.

Agar series combination mein ek member infinitely large ho jaaye (

C \to \infty

, effectively ek plain wire), to kya hoga?

Uska reciprocal

1/ C \to 0

sum se gayab ho jaata hai, isliye woh capacitor "disappear" ho jaata hai aur

C_{e q}

baaki members se set hota hai. Yeh ek filled bucket ke across short jaisa act karta hai.

Parallel bank mein, kya ek tiny capacitor add karna

C_{e q}

ko kaafi change karta hai?

Barely — woh sum mein sirf apni chhoti value add karta hai. Parallel sabse bade member se dominated hota hai; chhote wale near-negligible hain.

Series chain mein, kya ek huge capacitor add karna

C_{e q}

ko kaafi change karta hai?

Barely — uska reciprocal tiny hai, isliye woh sum se bahar ho jaata hai. Series sabse chhote member se dominated hota hai; bade wale near-negligible hain.

Agar do capacitors equal hain, to kya series aur parallel dono same answer dete hain?

Nahi. Equal

C

ke liye: parallel

2 C

deta hai, series

C /2

deta hai — ek factor of four ka antar. Members ki equality dono rules ko collapse nahi karti.

Kisi cheez ke saath "series" ya "parallel" mein akele ek capacitor ka

C_{e q}

kya hoga?

Bas

C

khud. Ek term ke saath sum ya reciprocal-sum wahi term return karta hai — dono formulas trivial case mein gracefully reduce ho jaate hain.

Active recall

Recall Two-line self-test

Ek reason batao kyun series charges equal hone chahiye — physics mein, formula se nahi.
Woh single member batao jo parallel bank ko dominate karta hai, aur woh jo series chain ko dominate karta hai.

Answers ::: (1) Do series capacitors ke beech ka middle conductor isolated hai aur net-neutral rehta hai, jo equal induced charge force karta hai. (2) Parallel sabse bade member se rule hota hai; series sabse chhote se.