Question

When circuit are created, electricians must take into consideration the electrical resistance to determine the appropriate gauge (diameter) of wire to use in a given situation. For example, household circuits are often wired with two different gauges of wire to accommodate different electrical needs such as light bulbs versus microwave ovens. The resistance of a wire varies directly as its length and inversely as the square of twice its radius. A wire 2.4 m long and 0.004 m in diameter has a resistance of 150 ohms. What is the diameter of a similar wire that has a resistance of 90 ohms and is 1.2 m long?

Answer 1

Answer:

0.00366 m

Step-by-step explanation:

We are given that

Length of wire=2.4 m

Diameter of wire=0.004 m

Radius of wire=$\frac{d}{2}=\frac{0.004}{2}=0.002 m$

Resistance of wire=150 ohm

We know that

Area of wire=$\pi r^2$

Resistance=$\rho\frac{L}{A^2}$

Substitute the values in the formula

$150=\rho\frac{2.4}{(3.14(0.002)^2}$

$\rho=\frac{150\times 3.14(0.002)^2}{2.4}$

$rho=7.85\times 10^{-4} ohm-m$

Length   of another wire=1.2 m

Resistance of wire=90 ohm

When wires are similar then resistivityremain same.

Therefore, resistivity of another wire=$7.85\times 10^{-4} ohm -m$

Substitute the values in the formula

$90=7.85\times 10^{-4}\times\frac{1.2}{3.14 r^2}$

$r=\sqrt{\frac{7.85\times 10^{-4}\times 1.2}{3.14\times 90}}=0.183\times 10^{-2}=1.83\times 10^{-3} m$

Diameter of wire=$2r=2\times 1.83\times 10^{-3}=3.66\times 10^{-3}=0.00366 m$