A length of wire has a radius of 3.00 x 10-3 m and a resistance of 0.200 ohms. When the potential
difference across the wire is 10.0 volts, the electron drift speed is found to be 2.98 x10-4 m/s. On the
basis of these data, calculate the density of free electrons in the wire.
J=nev,
where
J is the current density,
n is free electron density,
e=1.6•10⁻¹⁹C ,
drift velocity v= 2.98•10⁻⁴ m/s
J=U/RA,
where
U=10 V,
R=0.2 Ω,
A= πr²
n•e•v= U/R•π•r²
n= U/R•π•r²• e•v=...
To calculate the density of free electrons in the wire, we need to use the formula:
Density of free electrons = (current * cross-sectional area) / (electron charge * electron drift speed)
To find the current through the wire, we can use Ohm's Law:
Current = Voltage / Resistance
The cross-sectional area can be calculated using the formula:
Cross-sectional area = π * radius^2
The electron charge is a known constant, where its value is approximately 1.6 x 10^-19 C.
Now, let's calculate the density of free electrons step by step:
1. Calculate the cross-sectional area:
Cross-sectional area = π * (3.00 x 10^-3 m)^2 = 2.83 x 10^-5 m^2
2. Calculate the current:
Current = Voltage / Resistance
Current = 10.0 V / 0.200 Ω = 50.0 A
3. Calculate the electron density:
Density of free electrons = (Current * Cross-sectional area) / (Electron charge * Electron drift speed)
Density of free electrons = (50.0 A * 2.83 x 10^-5 m^2) / (1.6 x 10^-19 C * 2.98 x 10^-4 m/s)
Calculating the above expression will give you the density of free electrons in the wire.