On the basis of laws of series and parallel resistances, prove that the resistance of a conductor is directly proportional to its length and inversely proportional to its area of cross - section .
Answers
Given : basis of laws of series and parallel resistances
To find : prove that the resistance of a conductor is directly proportional to its length and inversely proportional to its area of cross section
Solution:
law of parallel resistances
The voltage across each resistor in parallel is same
=> I = V/R & I₁ = V/R₁ I₂ = V/R₂ ........... Iₙ = V/Rₙ
I = I₁ + I₂ + .............. + Iₙ
=> V/R = V/R₁ + V/R₂ +.......... ......... + V/Rₙ
Hence 1/R = 1/R₁ + 1/R₂ +............... 1/Rₙ
law of series resistances
Resistors in Series carry the same current,
V = V₁ + V₂ +................ + Vₙ
=> IR = IR₁ + IR₂ +............ + IRₙ
=> R = R₁ + R₂ +............ + Rₙ
Let say Two Resistance are added in length then these resistance are in series
=> Rtotal = R + R
=> RTotal = 2R
for R length was L
Then for 2L Length Resistance is 2R
2R/R = 2L/L
=> 2 = 2
=> R ∝ L
Hence resistance of a conductor is directly proportional to its length
Now if resistance are added in parallel their Cross Section area getting increased
for R crosssection Area was A
R total = 1/R + 1/R = R/2
Area = A + A = 2A
RA = (R/2)2A
=> RA = RA
=> R ∝ 1/A
Hence resistance of a conductor is inversely proportional to its area of cross section
R ∝ L
R ∝ 1/A
R ∝ L/A
R = ρ L/A
QED
Proved
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