Calculate equivalent resistance in parallel
Scenario
Agar ek nadi ke paani ko `3` alag-alag naharon (canals) mein baant diya jaye, toh kya hoga?
Paani (current) bat jayega, par shuru aur khatam hone ka level (voltage) same rahega.
Mental model
Parallel connection ka matlab hai raste ka batna. Jitne zyada raste khulenge, traffic (resistance) utna kam hoga.
Jab resistors ke dono ends ek dusre se jude hote hain, toh unke across potential difference (voltage) bilkul same rehta hai. Lekin total current un rasto mein apni-apni resistance ke hisab se bant jata hai.
Explanation
Parallel circuit mein, sabhi resistors battery ke dono terminals ke across directly jude hote hain. Iska matlab hai ki `V` (voltage) sab mein same hota hai.
Lekin main line se aane wala current (`I`) bat jata hai:
I = I_1 + I_2 + I_3
Ohm's law se `I = V / R` hota hai. Isliye:
V / R_p = V / R_1 + V / R_2 + V / R_3
Jab hum `V` ko common le kar cancel karte hain, toh equivalent resistance ka formula banta hai:
1 / R_p = 1 / R_1 + 1 / R_2 + 1 / R_3
Yahan inverse addition karna zaroori hai.
Ek khaas baat: Parallel mein laga equivalent resistance humesha circuit ke sabse chhote resistance se bhi chhota hota hai. Gharon mein saari wiring parallel mein hi hoti hai, taaki ek appliance band hone par baaki sab chalte rahein.
Worked example
Problem: Judge the equivalent resistance when `1 Ω` and `10^6 Ω` are connected in parallel.
- Parallel combination ka rule yaad karte hain.
- Conceptually, parallel me `R_p` sabse chote resistor se bhi chota hota hai.
Answer: Less than `1 Ω`
Problem: An electric lamp of `100 Ω`, a toaster of `50 Ω`, and a water filter of `500 Ω` are connected in parallel to a `220 V` source. What is the resistance of an iron connected to the same source that takes as much current as all three appliances, and what is the current through it?
- Teeno appliances parallel mein hain, unka equivalent `R_p` nikalte hain.
- LCM `500` lekar add karte hain.
- `R_p` ke liye flip karte hain.
- Total current `I = V / R_p` use karke nikalte hain.
Answer: Resistance = `31.25 Ω`, Current = `7.04 A`
Key points
- Constant Voltage: The voltage remains the same across all components in a parallel circuit.
- Current Divides: The total current is the sum of branch currents (`I = I_1 + I_2 + I_3`).
- Formula: `1 / R_p = 1 / R_1 + 1 / R_2 + 1 / R_3`
Common mistakes
- Forgetting to take the reciprocal: `1 / R_p` nikal kar wahi answer chhod dete hain. `R_p` nikalne ke liye fraction ko ulta karna hamesha yaad rakho.
Recall questions
- In a parallel circuit, what physical quantity remains the same across all branches?
- Is the equivalent resistance in parallel greater or less than the individual resistances?
- What remains constant across all resistors in a parallel circuit?
Questions & answers
What are the advantages of connecting electrical devices in parallel with the battery instead of connecting them in series?
1. If one fails, others keep working. 2. Each gets full voltage. 3. Total resistance is lower, providing more current.
Approach: List points comparing parallel and series.
Calculate the equivalent resistance of two `4 Ω` resistors in parallel.
`2 Ω`
Approach: `1 / R_p = 1 / 4 + 1 / 4 = 2 / 4 = 1 / 2`. So `R_p = 2 Ω`.
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