Series and parallel capacitors — derivations

1.8.12 · Physics › Electromagnetism

WHAT we are doing

Hum capacitors ke ek network ko ek single equivalent capacitor $C_{e q}$ se replace karte hain jo, jab same battery se connect ho, same total charge same total voltage pe draw kare. Yahi equivalent ki definition hai: $C_{e q} \equiv \frac{Q _{t o t a l}}{V _{t o t a l}}$

Parallel — derivation from scratch

HOW (step by step):

Har ek pe same voltage: $V_{1} = V_{2} = V_{3} = V$ . Yeh step kyun? Yeh same nodes share karte hain → equipotential wires.
Har ek pe charge: $Q_{1} = C_{1} V, Q_{2} = C_{2} V, Q_{3} = C_{3} V$ . Yeh step kyun? $Q = C V$ har capacitor pe alag se apply karo.
Battery total charge supply karti hai = sabhi branches ko deliver hua sum: $Q = Q_{1} + Q_{2} + Q_{3}$ Yeh step kyun? Node A pe charge conservation: battery se nikla charge branches mein split hota hai.
Substitute karo: $Q = C_{1} V + C_{2} V + C_{3} V = (C_{1} + C_{2} + C_{3}) V$
$Q = C_{e q} V$ se compare karo:

Series — derivation from scratch

HOW (step by step):

Har ek pe same charge: $Q_{1} = Q_{2} = Q_{3} = Q$ . Yeh step kyun? Isolated middle conductors pe charge induction (net-neutral rehna zaroori hai).
Har ek pe voltage: $V_{1} = \frac{Q}{C _{1}}, V_{2} = \frac{Q}{C _{2}}, V_{3} = \frac{Q}{C _{3}}$ . Yeh step kyun? Har ek ke liye $Q = C V$ rearrange karo.
Battery voltage = drops ka sum (Kirchhoff's voltage law loop ke around): $V = V_{1} + V_{2} + V_{3}$ Yeh step kyun? Series mein potential drops add hokar total applied PD dete hain.
Substitute karo: $V = \frac{Q}{C _{1}} + \frac{Q}{C _{2}} + \frac{Q}{C _{3}} = Q (\frac{1}{C _{1}} + \frac{1}{C _{2}} + \frac{1}{C _{3}})$
Lekin $V = \frac{Q}{C _{e q}}$ hai, toh dono sides ko $Q$ se divide karo:

Figure — Series and parallel capacitors — derivations

Forecast-then-Verify

Worked examples

Common mistakes (Steel-manned)

Active recall

Recall Dekhne se pehle try karo

Parallel rule batao aur do lines mein derive karo.
Series rule batao aur physical reason batao ki charges equal kyun hote hain.
Series mein kaun sa capacitor bada voltage drop leta hai aur kyun?
Series $C_{e q}$ hamesha sabse chote member se kyun kam hota hai?

Recall Feynman: ek 12-saal ke bachche ko samjhao

Capacitors ko electric charge ke liye buckets socho.

Parallel = buckets ko ek tap ke neeche side by side rakho, sab same water level (voltage) tak bharte hain. Zyada buckets = total mein zyada paani hold kar sakte ho → capacity add hoti hai.
Series = buckets ko ek single pipe mein stack karo taaki paani ki same trickle har ek se guzre. Har chota bucket paani ko zyada mushkil banata hai (zyada pressure drop), toh unhe stack karna cheez ko bharna mushkil kar deta hai → total capacity neeche jaati hai. Isliye hum "mushkilon" ( $1/ C$ ) ko add karte hain, buckets ko nahi.

Parallel capacitors mein har ek pe same... hoti hai

voltage (potential difference).

Series capacitors mein har ek pe same... hota hai

charge

Q

Parallel mein capacitors ka formula

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

(add hote hain).

Series mein capacitors ka formula

\frac{1}{C _{e q}} = \sum_{i} \frac{1}{C _{i}}

Do-capacitor series shortcut

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

(product over sum).

Series equivalent sabse chote se bhi chota kyun hota hai?

Reciprocals add karne se

1/ C_{e q}

sirf badhta hai, toh

C_{e q}

har member se neeche chala jaata hai.

Series mein kaun sa capacitor bada voltage leta hai?

Chota wala, kyunki

V = Q / C \propto 1/ C

hai aur

Q

common hai.

Parallel mein kaun sa capacitor bada charge leta hai?

Bada wala, kyunki

Q = C V \propto C

hai aur

V

common hai.

Series charges equal hone ki physical reason

Isolated middle conductors net-neutral rehte hain, har plate pe same

Q

induce karte hain.

Capacitor rules vs resistor rules

Yeh swapped hain: series-capacitors parallel-resistors jaisa behave karte hain aur vice versa.

Connections

Capacitance and the relation Q = CV — woh single equation jis par har step tiki hai.
Energy stored in a capacitor — $U = \frac{1}{2} C V^{2}$ ; jab capacitors combine hote hain toh redistribute hoti hai.
Resistors in series and parallel — mirror-image algebra; compare aur contrast karo.
Kirchhoff's voltage law — series loop mein "voltages add" ko justify karta hai.
Charge conservation — parallel mein node pe "charges add" ko justify karta hai.
Dielectrics in capacitors — combine karne se pehle slab insert karna individual $C_{i}$ ko kaise change karta hai.