Answer to Question #287575 in Electrical Engineering for Saiteja

Question #287575

A current of 90 A is shared by three resistances in parallel. The wires are of the


same material and have their lengths in the ratio 2: 3: 4 and their cross-sectional areas


in the ratio 1: 2: 3. Determine current in each resistance

1
Expert's answer
2022-01-17T08:00:04-0500

The voltage across all resistances is the same:


V=I1R1=I2R2=I3R3.V=I_1R_1=I_2R_2=I_3R_3.


By Kirchhoff's current law:


I1+I2+I3=90 A. VR1+VR2+VR3=90 A.I_1+I_2+I_3=90\text{ A}.\\\space\\ \dfrac V{R_1}+\dfrac V{R_2}+\dfrac V{R_3}=90\text{ A}.


The resistance of each resistor is


R1=ρL1πr12, R2=ρL2πr22=ρ(32L1)π(2r1)2=38R1, R3=ρL3πr32=ρ(2L1)π(3r1)2=29R1.R_1=\dfrac{\rho L_1}{\pi r_1^2},\\\space\\ R_2=\dfrac{\rho L_2}{\pi r_2^2}=\dfrac{\rho (\frac 32L_1)}{\pi (2r_1)^2}=\dfrac 38 R_1,\\\space\\ R_3=\dfrac{\rho L_3}{\pi r_3^2}=\dfrac{\rho (2L_1)}{\pi (3r_1)^2}=\dfrac 29 R_1.


Since I1R1=I2R2=I3R3,I_1R_1=I_2R_2=I_3R_3,


I1=38I2=29I3, I1+83I1+92I1=90, I1=11 A,I2=29 A,I3=50 A.I_1=\dfrac 38I_2=\dfrac29I_3,\\\space\\ I_1+\dfrac 83 I_1+\dfrac 92I_1=90,\\\space\\ I_1=11\text{ A},\\ I_2=29\text{ A},\\ I_3=50\text{ A}.


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