Apply resistivity formula
Scenario
Alag-alag companies ke wires same size ke hote hain, phir bhi ek jaldi garam ho jata hai aur dusra nahi.
Size same hone par unki aisi kaunsi internal property hai jo alag hoti hai?
Mental model
Resistivity material ka apna personal DNA (inherent property) hai. Wire moti ho ya patli, material same toh resistivity same.
Jab hum `R ∝ L` aur `R ∝ 1/A` ko jodne hain, toh equation banti hai `R = ρ * (L / A)`. Ye 'rho' (`ρ`) resistivity hai, jo sirf is baat par depend karti hai ki cheez bani kiski hai (jaise copper, iron, ya glass).
Explanation
Resistance (`R`) ka combination formula hai `R = ρ * (L / A)`, jahan `ρ` (rho) ek constant hai jise electrical resistivity kehte hain.
Resistivity ek fundamental property hai. Iska SI unit Ohm-meter (`Ω·m`) hota hai. Metals aur alloys ki resistivity bahut kam hoti hai (`10^-8` to `10^-6 Ω·m`), isliye wo good conductors hote hain. Rubber aur glass jaise insulators ki resistivity bahut zyada hoti hai.
Ek important point: Alloys (jaise Nichrome) ki resistivity unke constituent metals se zyada hoti hai, aur wo high temperature par jaldi oxidize (burn) nahi hote. Isiliye electric iron, toasters wagaira ki heating coils alloys ki banti hain, pure metals ki nahi.
Exam numericals mein aksar wire ka diameter `d` diya jata hai. Aise cases mein area nikalne ke liye `A = π * (d / 2)^2` use karte hain, aur dimensions ko meters mein convert karna bahut zaroori hai.
Worked example
Problem: A wire of given material having length `l` and area of cross-section `A` has a resistance of `4 ohm`. What would be the resistance of another wire of the same material having length `l/2` and area of cross-section `2A`?
- Pehli wire ka formula set up karte hain.
- Doosri wire ke naye dimensions put karte hain (material same hai, toh `ρ` same rahega).
- Equation ko simplify karte hain.
- Purani resistance ki value substitute karte hain.
Answer: `1 ohm`
Problem: Resistance of a metal wire of length `1 m` is `26 ohm`. If the diameter of the wire is `0.3 mm`, find the resistivity of the metal.
- Diameter ko meters mein convert karte hain.
- Area of cross section nikalte hain `A = π * (d / 2)^2`.
- Resistivity formula ko rearrange karke `ρ = (R * A) / L` use karte hain.
- Calculate karke final resistivity nikalte hain.
Answer: `1.83 * 10^-6 ohm-m`
Key points
- Resistivity Formula: `R = ρ * (L / A)`. Rearranging gives `ρ = (R * A) / L`.
- SI Unit: The SI unit of resistivity is Ohm-meter (`Ω·m`).
- Area from Diameter: If diameter `d` is given, `A = π * (d / 2)^2`.
Common mistakes
- Forgetting to convert mm to m: Area nikalte time diameter ko `mm` se meters mein convert karna bhool jate hain. Isse answer directly `1,000,000` times galat aata hai.
- Using diameter instead of radius: `A = π * r^2` ki jagah `A = π * d^2` use kar lete hain. Radius ke liye hamesha diameter ko `2` se divide karo.
Recall questions
- What is the SI unit of electrical resistivity?
- Does resistivity depend on the length of the wire?
- What is the SI unit of resistivity?
Questions & answers
Why are coils of electric toasters and electric irons made of an alloy rather than a pure metal?
Because alloys have higher resistivity than pure metals and do not oxidize (burn) readily at high temperatures.
Approach: Mention both resistivity and resistance to oxidation.
A wire of resistance `10 Ω` is doubled in length. What happens to its resistivity?
Resistivity remains unchanged.
Approach: Resistivity is a material property and does not change with physical dimensions like length.
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