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Chapter 18: Problem 15

A current of \(10.0 \mathrm{A}\) is carried by a copper wire of diameter $1.00 \mathrm{mm} .\( If the density of the conduction electrons is \)8.47 \times 10^{28} \mathrm{m}^{-3},$ how long does it take for a conduction electron to move \(1.00 \mathrm{m}\) along the wire?

Short Answer

Expert verified
Answer: To determine the time it takes for a conduction electron to move 1 meter along the copper wire, we must first calculate the cross-sectional area of the wire, find the drift speed of electrons, then calculate the time taken for an electron to move 1 meter along the wire. After performing these calculations, we find that the time, t = (1.00 m) / v, where v is the drift speed obtained from Step 2. Replace v with the calculated drift speed to find the time, t, for an electron to move 1.00 m along the copper wire.

Step by step solution

01

Calculate the cross-sectional area of the wire.

We're given the diameter of the copper wire, \(d=1.00\,\text{mm}.\) To find the cross-sectional area, first convert the diameter to meters: \(d = 1.00 \times 10^{-3}\,\text{m}\). Then, we can use the formula for the area of a circle: \(A = \pi (\frac{d}{2})^2\). $$ A = \pi (\frac{1.00\times 10^{-3}\,\text{m}}{2})^2 $$
02

Find the drift speed of electrons.

The formula for current is given by \(I=nAev,\) where \(I\) is the current, \(n\) is the density of conduction electrons, \(A\) is the cross-sectional area, \(e\) is the charge of an electron (\(1.60\times 10^{-19}\,\text{C}\)), and \(v\) is the drift speed. Rearrange the formula to make the drift speed, \(v\), the subject: $$ v = \frac{I}{nAe} $$ We are given the current, \(I = 10.0\,\text{A}\), and the density of conduction electrons, \(n = 8.47\times 10^{28}\,\text{m}^{-3}\). Plugging in the values, we find the drift speed of electrons: $$ v = \frac{10.0\,\text{A}}{(8.47\times 10^{28}\,\text{m}^{-3})\times (\pi (\frac{1.00\times 10^{-3}\,\text{m}}{2})^2) \times (1.60\times 10^{-19}\,\text{C})} $$
03

Calculate the time taken for an electron to move 1 meter along the wire.

We found the drift speed, \(v\), in the previous step. To find the time taken for an electron to move 1 meter along the wire, we use the formula \(t = \frac{d}{v}\), where \(t\) is the time, \(d\) is the distance, and \(v\) is the drift speed. We want to find the time it takes for an electron to move \(1.00\,\text{m}\) along the wire: $$ t = \frac{1.00\,\text{m}}{v} $$ Substitute the drift speed, \(v\), from Step 2, and find the value of \(t\). This will give us the time it takes for a conduction electron to move \(1.00\,\text{m}\) along the copper wire.

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