Question

An aluminum wire 5 m long and 2 mm diameter is connected in parallel with a copper wire 3m long.The total current is 4A and that is the aluminum wire is 2.5A.find the diameter of the copper wire.The resistivity of copper wire being 1.7molem and that of aluminum taken as 26nole ohms m.

Answer 1

Solution.

$l_a=5m;$

$d_a=2mm=0.002m;$

$\rho_a=2.6\sdot 10^{-8}Om\sdot m;$

$l_c=3m;$

$\rho_c=1.7\sdot10^{-8} Om\sdot m;$

$I=4A;$

$I_a=2.5A;$

$U_a=U_c\implies I_aR_a=I_cR_c;$

$R_a=\rho_a\dfrac{l_a}{S_a};$ $S_a=\pi\dfrac{d^2}{4};$

$S_a=\dfrac{3.14\sdot0.002^2}{4}=3.1\sdot 10^{-6}m^2;$

$R_a=2.6\sdot10^{-8}\dfrac{5}{3.1\sdot10^{-6}}=4.2\sdot 10^{-2}Om;$

$R_c=\dfrac{I_aR_a}{I_c};$

$I_c=I-I_a=4-2.5=1.5A;$

$R_c=\dfrac{2.5\sdot 4.2\sdot 10^{-2}}{1.5}=7\sdot 10^{-2}Om;$

$R_c=\rho_c\dfrac{l_c}{S_c}\implies S_c=\dfrac{\rho_cl_c}{R_c};$

$S_c=\dfrac{1.7\sdot 10^{-8}\sdot3}{7\sdot 10^{-2}}=0.7\sdot 10^{-6} m^2;$

$S_c=\pi\dfrac{d_c^2}{4}\implies d_c=2\sqrt{\dfrac{S_c}{\pi}};$

$d_c=2\sqrt{\dfrac{0.7\sdot10^{-6}}{3.14}}=1\sdot 10^{-3}m=1mm;$

Answer: $d_c=1mm.$