A wire is made from four different sections of the same metal, all sections bein

Question

A wire is made from four different sections of the same metal, all sections being cylin- drical. The lengths and diameters of the four sections are
L1=4cm   d1=4mm

L2=3cm   d2=2mm

L3=2cm   d3=3mm

L4=8cm   d4=5mm
The resistivity of the metal is 3.8 × 108 m. A current is passed through the wire. The current in, the resistance of, the voltage drop across, and the power dissipated in section x are, respectively, Ix, Rx, Vx, and Px.
(a) What is the ratio I1 :I2 :I3 :I4 ?

(b) What is the ratio R1 :R2 :R3 :R4 ?
(c) What is the ratio V1 :V2 :V3 :V4 ?
(d) What is the ratio P1 :P2 :P3 :P4 ?

(e) If the total power dissipated in the wire is 100 W, what is the power dissipated in the third section?

(f) What is the current in the wire?

Explanation / Answer

R = rho*l/A

Where rho is the resistivity

l is the length

A is the area of cross section

R1 = 1.21*10^-4 ohms

R2 = 3.63*10^-4 ohms

R3 = 1.08*10^-4 ohms

R4 = 1.55*10^-4 ohms

a) 1:1:1:1 as in series combination current is constant

B) 1.12:3.36:1:1.43

c) Same as ratio of resistance as V is directly proportional to resistance when current is same

d) Same as (b) as P = I^2R

e) Using

P = I^2R

Where R is total resistance = 7.47*10^-4 ohms

I = 365.9 A

Hence power dissipated by third resistor = 14.46 W

f) 365.9A

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