Finding the Density of Charge Carriers in a Liquid-Filled Pipe

Q1. A liquid-filled pipe contains Ca2+ ions (each ion has a net charge of +2e, where e = 1.6 × 10-19 C) and has a radius of 2.00 mm. When the current flowing along the pipe is 3.14 × 10-4 A, the carrier drift velocity is 5.00 × 10-5 m/s. What is the density of Ca2+ ions in the liquid-filled pipe?

Background

Topic: Current and Drift Speed

This question tests your understanding of how electric current relates to the density of charge carriers, their charge, drift velocity, and the cross-sectional area of the conductor. It is a classic application of the formula for current in terms of microscopic properties.

Key Terms and Formulas

• Current ($I$): The rate of flow of charge.

• Charge carrier density ($n$): Number of charge carriers per unit volume.

• Drift velocity ($v_d$): Average velocity of charge carriers due to an electric field.

• Charge of carrier ($q$): For Ca2+, $q = 2e$.

• Cross-sectional area ($A$): For a pipe of radius $r$, $A = \pi r^2$.

Key formula:

$I = n q v_d A$

Step-by-Step Guidance

1. Identify the known values: $I = 3.14 \times 10^{-4}$ A, $r = 2.00$ mm $= 2.00 \times 10^{-3}$ m, $v_d = 5.00 \times 10^{-5}$ m/s, $q = 2e = 2 \times 1.6 \times 10^{-19}$ C.

2. Calculate the cross-sectional area of the pipe: $A = \pi r^2$.

3. Write the formula for current in terms of charge carrier density: $I = n q v_d A$.

4. Rearrange the formula to solve for $n$: $n = \frac{I}{q v_d A}$.

5. Plug in the values for $I$, $q$, $v_d$, and $A$ to set up the calculation for $n$. Do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: $1.25 \times 10^{20}$ m-3

$A = \pi (2.00 \times 10^{-3})^2 = 1.26 \times 10^{-5}$ m2 $q = 2 \times 1.6 \times 10^{-19} = 3.2 \times 10^{-19}$ C $n = \frac{3.14 \times 10^{-4}}{(3.2 \times 10^{-19})(5.00 \times 10^{-5})(1.26 \times 10^{-5})} \approx 1.25 \times 10^{20}$ m-3

This is the density of Ca2+ ions in the pipe, found by relating the macroscopic current to the microscopic properties of the charge carriers.