Question

A uniform wire of resistance 12 $\Omega$ is cut into three pieces so that the ratio of the resistances R₁ : R₂ : R₃ = 1 : 2 : 3 and the three pieces are connected to form a triangle across which a cell of emf 8 V and internal resistance 1 W is connected as shown. Calculate the current through each part of the circuit.

Answer 1

ʳᵃ​​=1⃣3⃣​

ˢᵗᵉᵖ-ᵇʸ-ˢᵗᵉᵖ ᵉˣᵖˡᵃⁿᵃᵗⁱᵒⁿ:

ʰᵉʸ ᵐᵃᵗᵉ,


ᵍⁱᵛᵉⁿ,

ᵗʷᵒ ᵖⁱᵉᶜᵉˢ ᵒᶠ ᵐᵉᵗᵃˡ ᵃʳᵉ ᵃ ᵃⁿᵈ ᵇ.

ᵗʰᵉ ʳᵃᵗⁱᵒ ᵒᶠ ˡᵉⁿᵍᵗʰ ᵒᶠ ᵖⁱᵉᶜᵉˢ ᵃ ᵃⁿᵈ ᵇ ⁱˢ 1⃣ : 2⃣ ᵃⁿᵈ ʳᵃᵗⁱᵒ ᵒᶠ ʳᵃᵈⁱᵘˢ ᵒᶠ ᵖⁱᵉᶜᵉˢ ᵃ ᵃⁿᵈ ᵇ ⁱˢ 1⃣ : 2⃣.

= ˡ_{ᵃ} / ᵃ_{ᵇ} = 1⃣/2⃣ˡᵃ​/ᵃᵇ​=1⃣/2⃣  ᵃⁿᵈ  ʳ_{ᵃ} / ʳ_{ᵇ} = 1⃣/2⃣ʳᵃ​/ʳᵇ​=1⃣/2⃣


ʷᵉ ᵏⁿᵒʷ ᵗʰᵃᵗ ʳ = ᵖ\ᶠʳᵃᶜ{ˡ}{ᵃ}ʳ=ᵖᵃˡ​

ʷʰᵉʳᵉ ʳ ⁱˢ ʳᵉˢⁱˢᵗᵃⁿᶜᵉ, ˡ ⁱˢ ˡᵉⁿᵍᵗʰ ᵒᶠ ʷⁱʳᵉ ᵃⁿᵈ ᵃ ⁱˢ ᶜʳᵒˢˢ ˢᵉᶜᵗⁱᵒⁿᵃˡ ᵃʳᵉᵃ ᵒᶠ ʷⁱʳᵉ.


ˢᵒ, ʳ_{ᵃ} = \ᶠʳᵃᶜ{ˡ_{ᵃ}}{\ᵖⁱ ʳ^{2⃣}_{ᵃ} }ʳᵃ​=πʳᵃ2⃣​ˡᵃ​​ ᵃⁿᵈ ʳ_{ᵇ} = \ᶠʳᵃᶜ{ˡ_{ᵇ}}{\ᵖⁱ ʳ^{2⃣}_{ᵇ} }ʳᵇ​=πʳᵇ2⃣​ˡᵇ​​


ʷᵉ ᵍᵉᵗ,

\ᶠʳᵃᶜ{ʳ_{ᵃ}}{ʳ_{ᵇ}}=\ᶠʳᵃᶜ{ˡ_{ᵃ} ʳ^2⃣_{ᵇ}}{ˡ_{ᵇ}ʳ^2⃣_{ᵃ}}ʳᵇ​ʳᵃ​​=ˡᵇ​ʳᵃ2⃣​ˡᵃ​ʳᵇ2⃣​​

= \ᶠʳᵃᶜ{1⃣*2⃣^2⃣}{2⃣*1⃣^2⃣} = 2⃣=2⃣∗1⃣2⃣1⃣∗2⃣2⃣​=2⃣


ʷʰᵉⁿ ᵇᵒᵗʰ ᵖⁱᵉᶜᵉˢ ᵃʳᵉ ᶜᵒⁿⁿᵉᶜᵗᵉᵈ ⁱⁿ ᵖᵃʳᵃˡˡᵉˡ.

ʳ_{ᵉզ} = \ᶠʳᵃᶜ{ʳ_{ᵃ}ʳ_{ᵇ}}{ʳ_{ᵃ}+ ʳ_{ᵇ}} =\ᶠʳᵃᶜ{1⃣}{3⃣} ʳ_{ᵃ}ʳᵉզ​=ʳᵃ​+ʳᵇ​ʳᵃ​ʳᵇ​​=3⃣1⃣​ʳᵃ​


ˢᵒ,

\ᶠʳᵃᶜ{ʳ_{ᵃ}}{ʳ_{ᵉզ}} = \ᶠʳᵃᶜ{3⃣}{1⃣}ʳᵉզ​ʳᵃ​​=1⃣3⃣​