Question

A copper wire has a diameter \(d_{\mathrm{Cu}}=0.0500 \mathrm{~cm}\) is \(3.00 \mathrm{~m}\) long, and has a density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). As shown in the figure, the copper wire is attached to an equal length of aluminum wire with a diameter \(d_{\mathrm{A} \mathrm{I}}=0.0100 \mathrm{~cm}\) and density of charge carriers of \(6.02 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). A current of 0.400 A flows through the copper wire. a) What is the ratio of the current densities in the two wires, \(J_{\mathrm{Cu}} / J_{\mathrm{Al}} ?\) b) What is the ratio of the drift velocities in the two wires, \(v_{\mathrm{d}-\mathrm{Cu}} / v_{\mathrm{d}-\mathrm{Al}} ?\)

Short answer

Using the information provided and the geometrical properties of the wires, we calculated the current density and drift velocity for both the copper and aluminum wires. We found that the ratio of the current densities in the two wires is 0.04 and the ratio of drift velocities in the two wires is 5.

Step-by-step solution

Define the given parameters for the copper and aluminum wires.

We are given the following: For Copper wire: \(d_{Cu} = 0.0500\ cm\) Length = \(3.00\ m\) Density of charge carriers = \(8.50 \times 10^{28}\ electrons/m^3\) For Aluminum wire: \(d_{Al} = 0.0100\ cm\) Length = \(3.00\ m\) Density of charge carriers = \(6.02 \times 10^{28}\ electrons/m^3\) Current flowing through copper wire = \(I_{Cu} = 0.400\ A\)

Calculate cross-sectional areas for the copper and aluminum wires.

To find the cross-sectional area of each wire, we use the following formula: \( A_{wire} = \pi r^2 \) For Copper wire: \(A_{Cu} = \pi (\frac{d_{Cu}}{2})^2 = \pi (\frac{0.0500\ cm}{2})^2 = \pi (\frac{0.000500\ m}{2})^2 = \pi (0.000250)^2 m^2\) For Aluminum wire: \(A_{Al} = \pi (\frac{d_{Al}}{2})^2 = \pi (\frac{0.0100\ cm}{2})^2 = \pi (\frac{0.000100\ m}{2})^2 = \pi (0.000050)^2 m^2\)

Calculate current densities for the copper and aluminum wires.

The current density (J) is given by: \( J = \frac{I}{A} \) For Copper wire: \(J_{Cu} = \frac{I_{Cu}}{A_{Cu}} = \frac{0.400\ A}{\pi (0.000250)^2 m^2}\) For Aluminum wire: The current flowing through the Aluminum is the same as the current in the copper wire, hence \(I_{Al} = 0.400\ A\) \(J_{Al} = \frac{I_{Al}}{A_{Al}} = \frac{0.400\ A}{\pi (0.000050)^2 m^2}\)

Calculate the ratio of the current densities in the two wires.

Now, we can find the ratio of the current densities: \(J_{Cu}/J_{Al} = \frac{J_{Cu}}{J_{Al}} = \frac{\frac{0.400\ A}{\pi (0.000250)^2 m^2}}{\frac{0.400\ A}{\pi (0.000050)^2 m^2}} = \frac{(0.000050)^2}{(0.000250)^2} = \frac{0.00000250}{0.0000625} = 0.04\)

Calculate the drift velocities for the copper and aluminum wires.

The drift velocity (v) is given by: \( v = \frac{I}{nqA} \) For Copper wire: \(v_{d-Cu} = \frac{I_{Cu}}{n_{Cu}qA_{Cu}} = \frac{0.400\ A}{(8.50 \times 10^{28}\ electrons/m^3)(1.6 \times 10^{-19}\ C)(\pi (0.000250)^2 m^2)}\) For Aluminum wire: \(v_{d-Al} = \frac{I_{Al}}{n_{Al}qA_{Al}} = \frac{0.400\ A}{(6.02 \times 10^{28}\ electrons/m^3)(1.6 \times 10^{-19}\ C)(\pi (0.000050)^2 m^2)}\)

Calculate the ratio of the drift velocities in the two wires.

Now, we can find the ratio of the drift velocities: \(v_{d-Cu}/v_{d-Al} = \frac{v_{d-Cu}}{v_{d-Al}} = \frac{\frac{0.400\ A}{(8.50 \times 10^{28}\ electrons/m^3)(1.6 \times 10^{-19}\ C)(\pi (0.000250)^2 m^2)}}{\frac{0.400\ A}{(6.02 \times 10^{28}\ electrons/m^3)(1.6 \times 10^{-19}\ C)(\pi (0.000050)^2 m^2)}}\) \(v_{d-Cu}/v_{d-Al} = \frac{(0.000050)^2(6.02 \times 10^{28}\ electrons/m^3)}{(0.000250)^2(8.50 \times 10^{28}\ electrons/m^3)} = 5\) So, the ratio of the current densities in the two wires is \(0.04\) and the ratio of the drift velocities in the two wires is \(5\).

Key concepts

Current Density

Current density is an essential concept when dealing with electric currents in conductors. It represents the amount of current flowing per unit cross-sectional area of a material. Mathematically, it is expressed as:\[ J = \frac{I}{A} \]where \( J \) is the current density, \( I \) is the current, and \( A \) is the cross-sectional area. In our copper and aluminum wire exercise, understanding current density is crucial to determining the ratio of their current densities.If the current is constant, the difference in current densities between two materials usually depends on their cross-sectional areas. A smaller area will result in higher current density if the same amount of current passes through it.

• In the given case, the aluminum wire has a much smaller diameter than the copper wire.
• This difference leads to the aluminum wire having higher current density, as both wires carry the same current.

Appreciating current density helps one comprehend how electrons distribute across different materials under an electric field.

Drift Velocity

Drift velocity refers to the average velocity attained by charged particles, like electrons, as they move through a conductor under the influence of an electric field. The drift velocity \( v \) is related to current by the formula:\[ v = \frac{I}{nqA} \]Here, \( I \) is the current, \( n \) is the number density of charge carriers, \( q \) is the charge on each carrier, and \( A \) is the cross-sectional area of the wire.

• For our copper and aluminum wires, the drift velocity changes based on these parameters.
• Even though both materials have different carrier densities, it's the combination with area and constant current that dictates the drift speed.

Understanding drift velocity provides insight into how quickly charge carriers move within the wire, affecting how quickly electrical energy can be transferred within a circuit.

Charge Carriers

Charge carriers are particles, like electrons, that carry electric charge through a conductive material. Their density is crucial as it defines how much charge is available to move when a voltage is applied. In metals like copper and aluminum, electrons act as charge carriers.For our exercise:

• The copper wire has a charge carrier density of \( 8.50 \times 10^{28} \) electrons/m\(^3\), whereas the aluminum wire has \( 6.02 \times 10^{28} \) electrons/m\(^3\).
• This density affects both the drift velocity and overall resistance of the wire.

Higher charge carrier density typically means more electrons are available to deliver current. However, the actual "flow" efficiency also depends on the wire's cross-sectional area and additional properties like resistivity and material structure.

Copper and Aluminum Properties

Copper and aluminum are two metals widely used in electrical engineering due to their conductive properties. While both conduct electricity well, they have distinct characteristics:

• Copper has higher electrical conductivity than aluminum, meaning it can carry more current with less resistance.
• Aluminum is lighter and less expensive than copper, which makes it suitable for applications where weight and cost are concerns.
• Both metals have different densities of charge carriers, as seen in the exercise, affecting their efficiency in conducting electric current.

These material properties fundamentally impact calculations of current density and drift velocity. Hence, selecting the right material for specific applications involves balancing conductivity, weight, cost, and mechanical strength.