Current, Current Density, and Drift Velocity in a Composite Copper Conductor

Q1. Consider a copper conductor made of three sections: a 3 cm-long block with a 1.5 cm × 1.5 cm square cross-section, a 3 cm-long cylinder with a 1 cm diameter, and a plate that is 6 cm × 4 cm and 1 cm thick. A current of 20 A is passed through the conductor as shown.

Background

Topic: Electric Current, Current Density, and Drift Velocity in Conductors

This problem tests your understanding of how current, current density, and drift velocity relate in conductors of different shapes and cross-sectional areas. It also involves using material properties (like the number density of conduction electrons) and the concept of conductivity.

Key Terms and Formulas

• Current (I): The rate of flow of electric charge, measured in amperes (A).

• Current Density (J): The current per unit area, $J = \frac{I}{A}$, measured in A/m2.

• Number of Conduction Electrons (N_e): $N_e = n \cdot V$, where $n$ is the number density of conduction electrons and $V$ is the volume.

• Drift Velocity (v_d): The average velocity of conduction electrons due to an electric field, $I = n e v_d A$ or $v_d = \frac{J}{n e}$.

• Conductivity (\sigma): $J = \sigma E$, where $E$ is the electric field.

• Fundamental Charge (e): $e = 1.6 \times 10^{-19}$ C.

• Number Density for Copper (n): $n \approx 8.5 \times 10^{28}$ electrons/m3.

Step-by-Step Guidance

1. Part a: Calculate the number of conduction electrons, $N_e$, for each piece. - Find the volume $V$ of each section (block, rod, slab) using their dimensions. - Use $N_e = n \cdot V$ for each section, where $n$ is the number density for copper.

2. Part b: Determine the current $I$ through each piece. - Consider the fact that current is conserved in a series circuit (the same current flows through each section).

3. Part c: Calculate the current density $J$ for each piece. - Use $J = \frac{I}{A}$, where $A$ is the cross-sectional area of each section. - For the block and slab, $A$ is width × height; for the rod, $A = \pi r^2$.

4. Part d: Express the drift velocity $v_d$ in terms of $J$, $n$, and $e$. - Start from $I = n e v_d A$ and rearrange to solve for $v_d$ in terms of $J$, $n$, and $e$.

5. Part e: Use your result from part c to calculate the drift velocity $v_d$ for each piece. - Plug in the values for $J$, $n$, and $e$ for each section.

Try solving on your own before revealing the answer!