An aluminum wire of diameter 2.47 mm carries a current of 12.9 A. How long on average does it take an electron to move 10.5 m along the wire? Assume 3.50 conduction electrons per aluminum atom. The mass density of aluminum is 2.70 g/cm3 and its atomic mass is 27.0 g/mol.
One post of this question is sufficient.
Ne = (number density of electrons)
= 3.5 * (number density of aluminum atoms)
= 3.5 (electrons/atom)* (2.7 g/cm^3)/27 g/mole) *6.02*10^23 atoms/mol
Ne = 2.1*10*23 electrons/cm^3
For the velocity of the electrons,
V = I (Coulombs/s)/[e*(Area)*Ne]
If wire area in in cm^2, and Ne in electrons/cm^3, this will give you the velocity in cm/s
e is the electron charge, 1.60*10^19 coulombs/electron
To solve this problem, we need to first determine the number of electrons per unit volume in the aluminum wire. Then, we can calculate the drift speed of the electrons based on the current and the number of conduction electrons. Finally, we can find the average time it takes for an electron to travel a given distance.
Step 1: Find the number of electrons per unit volume
The number of conduction electrons per unit volume is given by:
\( n = \dfrac{N_A}{V} \),
where \( N_A \) is Avogadro's number (6.022 x 10^23), and \( V \) is the volume.
The volume \( V \) can be calculated using the formula for the volume of a cylinder:
\( V = \pi r^2 h \),
where \( r \) is the radius (half of the diameter), and \( h \) is the length of the wire.
Given the diameter of the wire is 2.47 mm, the radius \( r \) is:
\( r = \dfrac{2.47}{2} \) mm.
The length of the wire \( h \) is given as 10.5 m, but we need to convert it to mm:
\( h = 10.5 \times 1000 \) mm.
Now we can calculate the volume:
\( V = \pi \times \left(\dfrac{2.47}{2}\right)^2 \times 10.5 \times 1000 \) mm^3.
Step 2: Calculate the number of electrons per unit volume
Since we know there are 3.50 conduction electrons per aluminum atom, and the atomic mass of aluminum is 27.0 g/mol, we can calculate the number of atoms per unit volume using the mass density \( \rho \):
\( \rho = \dfrac{m}{V} \),
where \( m \) is the mass of the wire.
The mass of the wire is given by:
\( m = \pi \times \left(\dfrac{2.47}{2}\right)^2 \times 10.5 \times 1000 \times 2.70 \) g.
Now, we can calculate the number of atoms per unit volume:
\( N_{atoms} = \dfrac{\rho \times V}{M_{Al}} \),
where \( M_{Al} \) is the molar mass of aluminum.
Substituting the values, we can calculate \( N_{atoms} \).
Step 3: Calculate the drift speed of the electrons
The drift speed \( v_d \) is given by:
\( v_d = \dfrac{I}{n \times e} \),
where \( I \) is the current (12.9 A), \( n \) is the number of electrons per unit volume, and \( e \) is the charge of an electron.
Substituting the values, we can calculate \( v_d \).
Step 4: Calculate the average time for an electron to move the given distance
The average time \( t \) is given by:
\( t = \dfrac{d}{v_d} \),
where \( d \) is the given distance (10.5 m).
Substituting the values, we can calculate \( t \).
After performing all the calculations, we can find the average time it takes for an electron to move 10.5 m along the aluminum wire.