A current of 4 A flows in a copper wire 10 mm in diameter. The density of valence electrons in copper is roughly 9 × 1028 m−3.
Find the drift speed of these electrons. Answer in units of m/s.
To find the drift speed of electrons in a copper wire, we can use the formula:
v = I / (n * A * q)
Where:
v is the drift speed of electrons,
I is the current flowing through the wire,
n is the density of valence electrons in copper,
A is the cross-sectional area of the wire,
and q is the charge of an electron.
First, let's calculate the cross-sectional area (A) of the wire:
A = π * r^2
Where r is the radius of the wire. Since the diameter is given as 10 mm, we can calculate the radius:
r = 10 mm / 2 = 5 mm = 0.005 m
Substituting the radius into the formula for A:
A = π * (0.005 m)^2
Next, we need to determine the charge of an electron (q). The charge of an electron is approximately 1.6 × 10^-19 C (coulombs).
Now, let's substitute the given values and solve for the drift speed (v):
v = (4 A) / ((9 × 10^28 m^-3) * (π * (0.005 m)^2) * (1.6 × 10^-19 C))
After performing the calculations, we will have the drift speed in meters per second (m/s).
To find the drift speed of electrons in a copper wire, we can use the formula for drift velocity:
v_d = I / (n * A * q)
where:
- v_d is the drift speed of electrons,
- I is the current in the wire,
- n is the density of valence electrons in copper,
- A is the cross-sectional area of the wire, and
- q is the charge of an electron.
Let's plug in the given values:
I = 4 A
n = 9 × 10^28 m^(-3)
A = (π/4) * 10^(-2)^2 m^2 (since the wire has a circular cross-section)
q = 1.6 × 10^(-19) C (charge of an electron)
Calculating:
A = (π/4) * (0.01)^2 = 7.854 × 10^(-5) m^2
v_d = (4) / ((9 × 10^28) * (7.854 × 10^(-5)) * (1.6 × 10^(-19)))
Simplifying:
v_d ≈ 3.20 × 10^(-4) m/s
Therefore, the drift speed of electrons in the copper wire is approximately 3.20 × 10^(-4) m/s.