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Chapter 18: Problem 16

A silver wire of diameter 1.0 mm carries a current of \(150 \mathrm{mA} .\) The density of conduction electrons in silver is $5.8 \times 10^{28} \mathrm{m}^{-3} .$ How long (on average) does it take for a conduction electron to move \(1.0 \mathrm{cm}\) along the wire?

Short Answer

Expert verified
Answer: It takes approximately 400 seconds.

Step by step solution

01

Calculate the cross-sectional area of the wire.

First, we need to find the area of the wire using its diameter. The formula to calculate the area is A = πr^2, where A is the area, and r is the radius of the wire. Remember that 1.0 mm = 0.001 m. Radius (r) = diameter/2 = 0.001 m / 2 = 0.0005 m Area (A) = π × (0.0005 m)^2 = 7.85 x 10^-7 m^2
02

Calculate the number density of electrons.

The density of free electrons is given as 5.8 × 10^28 m^-3. This is the number of electrons per unit volume in the wire. We'll call this quantity n. Number density of electrons (n) = 5.8 × 10^28 m^-3
03

Find the current.

The current flowing through the wire is given as 150 mA. Convert it to amperes. Current (I) = 150 mA = 0.15 A
04

Calculate the drift velocity.

We can use the formula for current in terms of the drift velocity: I = n × e × A × v_d where I is the current, n is the number density of electrons, e is the charge of an electron (1.6 x 10^-19 C), A is the area, and v_d is the drift velocity. We need to solve for drift velocity (v_d). 0.15 A = (5.8 × 10^28 m^-3) × (1.6 × 10^-19 C) × (7.85 × 10^-7 m^2) × v_d Now, solve for v_d: v_d ≈ 2.5 × 10^-5 m/s
05

Calculate the time to drift 1.0 cm.

Finally, we can use the drift velocity to find the average time it takes an electron to drift 1.0 cm (0.01 m) along the wire. We'll use the formula: time = distance / drift velocity Average time (t_avg) = (1.0 cm) / (2.5 × 10^-5 m/s) = 0.01 m / (2.5 × 10^-5 m/s) ≈ 400 s So, on average, it takes a conduction electron about 400 seconds to move 1.0 cm along the silver wire.

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Most popular questions from this chapter

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