Question

You've made the finals of the Science Olympics! As one of your tasks, you're given 1.9 g of copper and asked to make a cylindrical wire, using all the metal, with a resistance of 1.8 Ω.

Part A What length ℓ will you choose for your wire? Express your answer in meters to two significant figures. Part B What diameter d will you choose for your wire? Express your answers in millimeters to two significant figures

Answer 1

Given that mass of copper wire = 1.9 gm

density of copper = 8.96 gm/cm^3

Now we know that

Mass = Density*Volume

Volume = Mass/Density

Volume = 1.9/8.96 = 0.21205 cm^3

Now Volume of cylindrical wire is given by:

Volume = pi*r^2*L = pi*d^2*L/4

(since radius, r = diameter/2 = d/2) So

pi*d^2*L/4 = 0.21205 cm^3 = 0.1016*10^-6 m^3

d^2*L = 4*0.21205*10^-6/pi = 2.70*10^-7

Now Also given that

Resistance of wire is

R = rho*L/A = 4*rho*L/(pi*d^2) = 1.8 ohm

rho = resistivity of copper wire = 1.72*10^-8 ohm-m, So

L/d^2 = 1.8*pi/(4*1.72*10^-8) = 8.22*10^7

Now from above two equations:

d^2*L = 2.70*10^-7

L/d^2 = 8.22*10^7

Multiply both equations:

L^2 = 2.70*10^-7*8.22*10^7

L = sqrt (2.70*8.22) = 4.711 m

In two significant figures

Length of wire = 4.7 m

Using above value:

d^2 = 2.70*10^-7/4.711 = 5.731267*10^-8

d = sqrt (5.731267*10^-8) = 0.24*10^-3 m

d = 0.24 mm = diameter of wire

Let me know if you've any query.