A copper wire has a diameter of 1.532 mm. What magnitude current flows when the drift velocity is 0.500 mm/s?
To find the current flowing through the copper wire, we can use the formula:
I = n*A*drift_velocity*q
where:
I is the current,
n is the number density of the copper (number of charge carriers per unit volume),
A is the cross-sectional area of the wire,
drift_velocity is the drift velocity of the electrons, and
q is the elementary charge, which is approximately 1.60 x 10^(-19) C.
First, we need to find the number density of the copper. The number density can be calculated using the formula:
n = density * (N_A / molar_mass)
where:
density is the density of the copper (approximately 8.96 g/cm^3),
N_A is Avogadro's number (approximately 6.02 x 10^23 atoms/mole), and
molar_mass is the molar mass of copper (approximately 63.5 g/mole).
n = (8.96 g/cm^3) * (6.02 x 10^23 atoms/mole) / (63.5 g/mole)
To make the units consistent, let's convert the density from g/cm^3 to kg/m^3:
density = 8.96 g/cm^3 * (1 kg/1000 g) * (100 cm/m)^3 = 8960 kg/m^3
Now we can find the number density:
n = (8960 kg/m^3) * (6.02 x 10^23 atoms/mole) / (63.5 g/mole * (1 kg/1000 g))
n = 8.48 x 10^28 atoms/m^3
Next, we need to find the cross-sectional area of the copper wire. The area can be calculated using the formula:
A = pi * (diameter / 2)^2
A = pi * (1.532 mm / 2)^2
To make the units consistent, let's convert the diameter from mm to meters:
diameter = 1.532 mm * (1 m / 1000 mm) = 0.001532 m
Now we can find the cross-sectional area:
A = pi * (0.001532 m / 2)^2
A = 1.843 x 10^(-6) m^2
Finally, we can calculate the current flowing through the copper wire:
I = n * A * drift_velocity * q
I = (8.48 x 10^28 atoms/m^3) * (1.843 x 10^(-6) m^2) * (0.5 mm/s * (1 m / 1000 mm)) * (1.6 x 10^(-19) C)
I = 12.9 A
The magnitude of the current flowing through the copper wire is approximately 12.9 A.
To determine the magnitude of the current flowing through a copper wire with a given diameter and drift velocity, we can use the equation:
I = nAvq
where:
I is the magnitude of the current
n is the number density of charge carriers
A is the cross-sectional area of the wire
v is the drift velocity of the charge carriers
q is the elementary charge
Let's calculate step by step:
1. Convert the wire diameter to radius:
Radius = Diameter / 2 = 1.532 mm / 2 = 0.766 mm = 0.766 × 10^-3 m
2. Calculate the cross-sectional area of the wire:
A = πr^2
= π × (0.766 × 10^-3)^2
= π × (0.588356 × 10^-6)
= 1.839 × 10^-6 m^2
3. Determine the number density of charge carriers for copper:
In copper, each atom contributes one free electron for conduction per atom. The number density of copper atoms can be approximated as 8.5 × 10^28 atoms/m^3.
4. Calculate the number density of charge carriers:
n = number density of copper atoms × number of free electrons per atom
= 8.5 × 10^28 atoms/m^3 × 1 electron/atom
= 8.5 × 10^28 electrons/m^3
5. Calculate the magnitude of the current:
We know that the drift velocity (v) = 0.500 mm/s = 0.500 × 10^-3 m/s
Let's assume the elementary charge (q) is 1.602 × 10^-19 C (Coulombs).
I = nAvq
= (8.5 × 10^28 electrons/m^3) × (1.839 × 10^-6 m^2) × (0.500 × 10^-3 m/s) × (1.602 × 10^-19 C)
= 12.6 A
Therefore, the magnitude of the current flowing through the copper wire is 12.6 Amperes (A).