A Cu wire of diameter 1 mm is carrying a current of 15 mA. What is the drift velocity of the conduction electrons? The drift mobility of Cu is 43.4x10-4 m2
V-1 s-1, and the conductivity of Cu is 5.9x107 Ω-1 m-1.
To find the drift velocity of the conduction electrons in the Cu wire, we can use the formula:
v = (I / nAe)
Where:
- v is the drift velocity
- I is the current
- n is the number density of conduction electrons
- A is the cross-sectional area of the wire
- e is the elementary charge
First, let's find the number density of conduction electrons (n):
- The drift mobility of Cu (μ) is given as 43.4 x 10^(-4) m^2 V^(-1) s^(-1). The formula to find the number density is:
n = (μ ρ) / e
Where:
- ρ is the resistivity (inverse of the conductivity)
- e is the elementary charge
For Cu, the resistivity (ρ) is given as 5.9 x 10^7 Ω^(-1) m^(-1), and the elementary charge (e) is approximately 1.6 x 10^(-19) C.
Plugging in the values, we get:
n = (43.4 x 10^(-4) m^2 V^(-1) s^(-1) * 5.9 x 10^7 Ω^(-1) m^(-1)) / (1.6 x 10^(-19) C)
Simplifying the expression, we get:
n = 16.11 x 10^(29) m^(-3)
Now, let's calculate the cross-sectional area of the wire (A):
- The diameter of the wire is given as 1 mm. To find the radius (r), divide the diameter by 2.
r = 1 mm / 2 = 0.5 mm = 0.5 x 10^(-3) m
- The cross-sectional area (A) is given by:
A = πr^2
Plugging in the values, we get:
A = π * (0.5 x 10^(-3) m)^2
Simplifying the expression, we get:
A ≈ 7.85 x 10^(-7) m^2
Now, substitute the calculated values into the drift velocity formula:
v = (I / nAe)
v = (15 mA / (16.11 x 10^29 m^(-3) * 7.85 x 10^(-7) m^2 * 1.6 x 10^(-19) C))
Simplifying the expression, we get:
v = 2.40 x 10^(-3) m/s
Therefore, the drift velocity of the conduction electrons in the Cu wire is approximately 2.40 x 10^(-3) m/s.