Question

What is the current density in an aluminum wire having a radius of \(1.00 \mathrm{~mm}\) and carrying a current of \(1.00 \mathrm{~mA}\) ? What is the drift speed of the electrons carrying this current? The density of aluminum is \(2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3},\) and 1 mole of aluminum has a mass of \(26.98 \mathrm{~g}\). There is one conduction electron per atom in aluminum.

Short answer

Based on the given information in the problem, the current density in the aluminum wire is approximately \(3.18\cdot10^2\, \text{A/m}^2\), and the drift speed of the electrons carrying this current is approximately \(3.29\cdot10^{-4}\, \text{m/s}\).

Step-by-step solution

Calculate the cross-sectional area of the wire

First, let's calculate the cross-sectional area \(A\) of the aluminum wire using its given radius \(r=1.00\,\text{mm}=1.00\cdot 10^{-3}\, \text{m}\). Since the wire is cylindrical, we can calculate its area using the formula \(A= \pi r^2\): $$ A = \pi r^2 = \pi (1.00\cdot10^{-3}\, \text{m})^2 = 3.14\cdot10^{-6}\, \text{m}^2. $$

Calculate the current density

Now that we have the cross-sectional area of the wire, we can calculate the current density \(J\) using the formula \(J=I/A\), where \(I=1.00\, \text{mA}=1.00\cdot10^{-3}\, \text{A}\): $$ J = \frac{I}{A} = \frac{1.00\cdot10^{-3}\, \text{A}}{3.14\cdot10^{-6}\, \text{m}^2} = 3.18\cdot10^2\, \frac{\text{A}}{\text{m}^2}. $$

Calculate the number of moles in a unit volume of aluminum

The next step is to find the charge density \(n\) of the aluminum wire. First, we need to calculate the number of moles in a unit volume of aluminum. We know the density of aluminum is \(\rho = 2.70\cdot10^3\, \text{kg/m}^3\), and 1 mole of aluminum has a mass of \(M = 26.98\, \text{g} = 26.98\cdot10^{-3}\, \text{kg}\): $$ \text{Number of moles per unit volume} = \frac{\rho}{M} = \frac{2.70\cdot10^3\, \text{kg/m}^3}{26.98\cdot10^{-3}\, \text{kg/mol}} = 1.00\cdot10^{5}\, \text{mol/m}^3. $$

Calculate the charge density

Since there is one conduction electron per atom in aluminum, the charge density \(n\) is equal to the number of moles per unit volume multiplied by Avogadro's number \(N_A = 6.022\cdot10^{23}\, \text{mol}^{-1}\): $$ n = (1.00\cdot10^{5}\, \text{mol/m}^3)\cdot(6.022\cdot10^{23}\, \text{mol}^{-1}) = 6.022\cdot10^{28}\, \text{m}^{-3}. $$

Calculate the drift speed of electrons

Finally, we can calculate the drift speed of the electrons \(v_d\) using the formula \(v_d = I/(neA)\), where \(e\) is the elementary charge \(e = 1.60\cdot10^{-19}\, \text{C}\): $$ v_d = \frac{I}{neA} = \frac{1.00\cdot10^{-3}\, \text{A}}{(6.022\cdot10^{28}\, \text{m}^{-3})(1.60\cdot10^{-19}\, \text{C})(3.14\cdot10^{-6}\, \text{m}^2)} = 3.29\cdot10^{-4}\, \frac{\text{m}}{\text{s}}. $$ So, the current density in the aluminum wire is \(3.18\cdot10^2\, \text{A/m}^2\), and the drift speed of the electrons carrying this current is \(3.29\cdot10^{-4}\, \text{m/s}\).

Key concepts

Drift Speed

The concept of drift speed is important in understanding how electrons move through a conductor. Drift speed is the average velocity that electrons attain due to an electric field. Despite the large number of electrons moving randomly, their net flow in the direction of the electric field is responsible for the current. In essence, drift speed is a measure of how quickly charge carriers, like electrons, are moving.
For conductors like aluminum, the drift speed is surprisingly slow compared to the speed of electromagnetic signals. During computations, drift speed can be determined using the formula \( v_d = \frac{I}{neA} \). Here:

• \( I \) is the current flowing through the conductor,
• \( n \) is the number density of electrons,
• \( e \) stands for the elementary charge, and
• \( A \) represents the cross-sectional area.

The calculated drift speed of electrons here helps us visualize how energy transfer can occur efficiently, despite the slow movement of individual electrons. This slow drift speed is a reminder of the collective effect of countless electrons contributing to the current.

Conductivity of Aluminum

To understand the conductivity of aluminum, we need to first recognize that conductivity is a measure of how easily electrons can flow through a material. Aluminum is known for its good conductivity, which makes it a popular choice in electrical applications.
Conductivity, represented by \( \sigma \), is linked to the current density \( J \) and the electric field \( E \) in the material by the relation \( J = \sigma E \). A material with high conductivity will allow a greater flow of electric current for a given electric field.
Aluminum's high conductivity is attributed to:

• Its crystal structure, which allows electrons to move freely,
• The presence of a large number of conduction electrons, and
• The metal's relatively low atomic mass, which reduces resistance to electron flow.

In the original exercise, the current density was calculated to show how much current passes through the wire influenced by its conductivity. This property helps in efficient power transmission and electrical uses of the metal.

Number Density of Electrons

The number density of electrons is crucial for understanding various electrical properties of a material, including current density and drift speed. This concept explains how many conduction electrons are available per unit volume in a material.
To calculate this, one needs to determine the number of atoms in a given volume and recognize that in metals like aluminum, each atom typically contributes one conduction electron. For aluminum, the number density can be calculated using its molar density and Avogadro's number.

• The density of aluminum and its molar mass tells us the number of moles per unit volume.
• Avogadro's number provides the number of atoms (and thus, conduction electrons) per mole.

Hence, the number density \( n \) is given by multiplying the number of moles per unit volume by Avogadro's number, resulting in a large value showing a high availability of conductive charge carriers.
Understanding the number density helps in calculating both the current density \( J \) and drift speed \( v_d \). The concept underscores the availability of electrons that facilitate electric current in conductive materials such as aluminum.