Question

(a) A steel power transmission line has a resistance of 0.0430 Ω/km. What is its mass per kilometer (in kg/km)? (Assume the density of steel is 7.8 ✕ 10³ kg/m³.)

  ________ kg/km

(b) What is the mass per kilometer (in kg/km) of an iron line having the same resistance? (Assume the density of iron is 7.8 ✕ 10³ kg/m³.)

_______ kg/km

Answer 1

Since, Resistance is given by,

R = rho*L/A

R/L = rho/A

For steel,

rho = resistivity of steel = 2.0*10^-7 ohm*m

given, R/L = 0.0430 ohm/km = 0.0430*10^-3 ohm/m

So, A = area of transmission line = rho/(R/L)

A = (2.0*10^-7)/(0.0430*10^-3)

A = 4.651*10^-3 m^2

Now, Mass(M) = *V

here, = density of steel = 7.8*10^3 kg/m^3

V = volume = area(A)*length(L)

then, M = *A*L

M/L = *A

M/L = (7.8*10^3)*(4.651*10^-3)

M/L = 36.279 kg/m

M/L = 36.279*10^3 kg/km

M/L = 36279 kg/km

b.)

For iron,

rho = resistivity of iron = 1.0*10^-7 ohm*m

given, R/L = 0.0460 ohm/km = 0.0460*10^-3 ohm/m

So, A = area of transmission line = rho/(R/L)

A = (1.0*10^-7)/(0.0430*10^-3)

A = 2.325*10^-3 m^2

Now, Mass(M) = *V

here, = density of iron = 7.8*10^3 kg/m^3

V = volume = area(A)*length(L)

then, M = *A*L

M/L = *A

M/L = (7.8*10^3)*(2.325*10^-3)

M/L = 18.139 kg/m

M/L = 18.139*10^3 kg/km

M/L = 18139 kg/km