MATHEMATICS and ELECTRICAL ENGINEERING: Joint and Combined Variation - Wire resistance, length, area

WIRE RESISTANCE is directly proportional to LENGTH and inversely proportional to AREA

R = k L/A


where:

R = wire resistance

k = constant

L = length of wire

A = cross-sectional area of wire



1. If 700 meters of a 4 mm-diameter wire has a resistance of 28 ohms, find the length of a 7 mm-dia wire of same resistance.

given:

L1 = 700 m

d1 = 4 mm

R1 = 28 ohms

d2 = 7 mm

R2 = 28 ohms = R1


find:

L2 = length of wire with 7 mm diameter having same resistance of 28 ohms


solution:


R = k L/A

k = R * A/L

R1 * A1/L1 = R2 * A2/L2

28 * (4^2 * pi/4)/700 = 28 * (7^2 * pi/4)/L2

16/700 = 49/L2

L2 = 700 * 49/16

L2 = 2144 m




2. By what factor will the wire resistance change if both the wire length and diameter are increased by 200% (doubled)?

given:

L1 = L

d1 = d

L2 = 2L

d2 = 2d


find:

change factor of wire resistance when wire length and wire diameter are both doubled


solution:


R = k L/A

k = R * A/L

R1 * A1/L1 = R2 * A2/L2

R1 * (d^2 * pi/4)/L = R2 * [(2d)^2 * pi/4]/2L

R1 * d^2 = R2 * (4 * d^2)/2

R1 = R2 * 2

R2 = (R1)/2

R2 = 0.5 R1  ---> the final wire resistance (R2) is REDUCED to HALF of the initial wire resistance (R1)