Question

Electrons flow through a 1.8-mm-diameter aluminum wire at 1.5×10 ^−4 m/s. How many electrons move through a cross section of the wire each day?

Answer 1

To calculate how many electrons move through the wire each day, calculate the wire's cross-sectional area, determine the number of charge carriers per unit volume, and compute the total charge flow per day. From this, use the electron charge to determine the total number of electrons.

Step-by-step explanation:

To determine how many electrons move through a cross section of the wire each day, we need to calculate the total charge that moves through that cross section and then use the charge of a single electron to find the number of electrons.

First, let's find the cross-sectional area (A) of the wire using the formula A = πr², where r is the radius of the wire.
Since the diameter of the wire is 1.8 mm, its radius r is 0.9 mm or 0.9×10⁻³ m.
Hence:

A = π(0.9×10⁻³ m)² = π(0.81×10⁻¶ m²) = π(0.81×10⁻¶ m²)

Next, the charge Q that moves through the wire per second (current, I) is given by I = nqAvd, with n as the number of charge carriers per unit volume, q as the charge of an electron, and vd as the drift velocity. Since vd is provided, we can rearrange this to solve for Q.

The number of charge carriers, n, can be found as follows:
Given the density (ρ) of aluminum is 2.70×10³ kg/m³ and its molar mass (M) is 26.98 g/mol, we can use Avogadro's number (NA) to find n:

n = (ρ/M) × NA

Thus, we calculate the total number of electrons passing through the cross-section in a day by multiplying the charge per second by the number of seconds in a day (86400 seconds) and then dividing by the charge of a single electron (q).

Finally, we reach an answer that gives us the quantity in terms of the number of electrons that pass through the cross-section of the wire in one day. This elaborate process applies knowledge from electrical and physical concepts to provide a concrete answer.

Answer 2

Approximately
`\approx 2.77 * 10^(21) \text { electrons/day }` electrons move through the cross-section of the aluminum wire each day.

To calculate the number of electrons that move through a cross-section of the wire each day, you can use the formula:

I= n ⋅ A ⋅ v ⋅ q

where:

I is the electric current,

n is the number of electrons per unit volume,

A is the cross-sectional area of the wire,

v is the drift velocity of the electrons,

q is the charge of an electron.

Calculate the cross-sectional area:

`\begin{equation}\begin{aligned}& r=(d)/(2)=\frac{1.8 \mathrm{~mm}}{2}=0.0009 \mathrm{~m} \\& A=\pi *(0.0009 \mathrm{~m})^2 \\& A \approx \pi *\left(8.1 * 10^(-7) \mathrm{~m}^2\right) \\& A \approx 2.02 * 10^(-6) \mathrm{~m}^2\end{aligned}\end{equation}`

Find the number of electrons per unit volume (n):

`\begin{equation}\text { For aluminum, } n \approx 8.5 * 10^(28) \text { electrons } / \mathrm{m}^3 \text {. }\end{equation}`

Calculate the current (I):

`\begin{aligned}& I=n \cdot A \cdot v \cdot q \\& I=\left(8.5 * 10^(28) \text { electrons } / \mathrm{m}^3\right) *\left(2.02 * 10^(-6) \mathrm{~m}^2\right) *\left(1.5 * 10^(-4) \mathrm{~m} / \mathrm{s}\right) * \\& \left(1.602 * 10^(-19) \mathrm{C}\right)\end{aligned}`

Calculating this expression gives I. Let's compute it:

`\begin{aligned}& I \approx\left(8.5 * 10^(28)\right) *\left(2.02 * 10^(-6)\right) *\left(1.5 * 10^(-4)\right) *\left(1.602 * 10^(-19)\right) \\& I \approx 5.14 * 10^(-3) \mathrm{~A}\end{aligned}`

Convert current to the number of electrons per second:

Number of electrons per second = I/e

Number of electrons per second =
`(5.14 * 10^(-3))/(1.602 * 10^(-19))`

Number of electrons per second
`\approx 3.21 * 10^(16) \text { electrons/second }`

Calculate the total number of electrons each day:

Number of electrons per day
`\approx 3.21 * 10^(16) * 86400`

Number of electrons per day
`\approx 2.77 * 10^(21) \text { electrons/day }`

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