A length of wire has a radius of 3.00 �~ 10-3 m and a resistance of 0.200 ƒ¶. When the potential
difference across the wire is 10.0 V, the electron drift speed is found to be 2.98 �~ 10-4 m/s. On the
basis of these data, calculate the density of free electrons in the wire.
To calculate the density of free electrons in the wire, we can use the formula:
n = ( I * A ) / (q * v_d)
Where:
n is the density of free electrons,
I is the current flowing through the wire,
A is the cross-sectional area of the wire,
q is the charge of an electron, and
v_d is the drift velocity of electrons.
First, let's calculate the current (I) using Ohm's Law:
I = V / R
Where:
V is the potential difference across the wire, and
R is the resistance of the wire.
Given V = 10.0 V and R = 0.200 Ω, we can find I:
I = 10.0 V / 0.200 Ω = 50 A
Next, let's calculate the cross-sectional area (A) of the wire using the formula:
A = π * r^2
Where:
r is the radius of the wire.
Given r = 3.00 × 10^-3 m, we can find A:
A = π * (3.00 × 10^-3 m)^2
Now, let's calculate the charge of an electron (q):
q = 1.6 × 10^-19 C (known value)
Finally, given the drift velocity (v_d) is 2.98 × 10^-4 m/s, we can calculate the density of free electrons (n):
n = (50 A * π * (3.00 × 10^-3 m)^2) / (1.6 × 10^-19 C * 2.98 × 10^-4 m/s)
Simplifying the equation will give you the density of free electrons in the wire.