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184_notes:examples:week6_drift_speed [2017/09/26 16:04] – [Solution] tallpaul | 184_notes:examples:week6_drift_speed [2018/02/03 22:24] – [Solution] tallpaul | ||
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=====Example: | =====Example: | ||
- | Suppose you have a two wires. Each has a current of $5 \text{ A}$. One is made of copper (Cu) and has radius $0.5 \text{ mm}$. The other is made of zinc (Zn) and has radius $0.1 \text{ mm}$. What is the drift speed of electrons in each wire? You may want to consult the table below. | + | Suppose you have a two wires. Each has a current of $5 \text{ A}$. One is made of copper (Cu) and has radius $0.5 \text{ mm}$. The other is made of zinc (Zn) and has radius $0.1 \text{ mm}$. What are the drift speeds |
- | {{ 184_notes: | + | {{ 184_notes: |
===Facts=== | ===Facts=== | ||
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* The zinc wire has $I=5 \text{ A}$, $r = 0.1 \text{ mm}$. | * The zinc wire has $I=5 \text{ A}$, $r = 0.1 \text{ mm}$. | ||
* The charge of an electron is $q=-1.6\cdot 10^{-19} \text{ C}$. | * The charge of an electron is $q=-1.6\cdot 10^{-19} \text{ C}$. | ||
+ | * Electron density of copper is $n_{\text{Cu}}=8.47\cdot 10^{22} \text{ cm}^{-3}$. | ||
+ | * Electron density of zinc is $n_{\text{Zn}}=13.2\cdot 10^{22} \text{ cm}^{-3}$. | ||
+ | * Electron current as $i=nAv_{avg}$. | ||
+ | * Current is $I=|q|i$. | ||
+ | * Units of current is charge per second. Electron current is electrons per second. We multiply by $q$ (the electron charge) to get charge per second. | ||
- | ===Lacking=== | + | ===Goal=== |
- | * Drift speed for both wires. | + | * Find the drift speed for both wires. |
- | * Electron charge density for both wires. | + | |
- | * Electron current for both wires. | + | |
- | * Cross-sectional area for both wires. | + | |
===Approximations & Assumptions=== | ===Approximations & Assumptions=== | ||
- | * The table is accurate for our wires. | ||
* The wires have circular cross-sections. | * The wires have circular cross-sections. | ||
- | * The wires do not experience any external | + | * Using the [[184_notes: |
===Representations=== | ===Representations=== | ||
* We represent electron current as $i=nAv_{avg}$. | * We represent electron current as $i=nAv_{avg}$. | ||
- | * We represent current as $I=qi$. Current is charge per second. Electron current is electrons per second. We multiply by $q$ (the electron charge) to get charge per second. | + | * We represent current as $I=|q|i$. Current is charge per second. Electron current is electrons per second. We multiply by $q$ (the electron charge) to get charge per second. |
+ | * | ||
====Solution==== | ====Solution==== | ||
- | We can look up electron density $n$ in the table. It is labeled as " | + | We can use the [[184_notes: |
+ | |||
+ | There is a lot going on in this problem, so let's make a plan. | ||
+ | |||
+ | <WRAP TIP> | ||
+ | === Plan === | ||
+ | We will do the following steps for each wire. | ||
+ | * Find the electron density of each material (see listed above, in Facts). | ||
+ | * Find the cross-sectional area of the wire. | ||
+ | * Find the electron current of each wire, using the given current. | ||
+ | * Use all the new information to find the drift speed. | ||
+ | </ | ||
To find the cross-sectional area of the wire, we just use the area of a circle. We know the radius, so this should be easy: $A=\pi r^2$. | To find the cross-sectional area of the wire, we just use the area of a circle. We know the radius, so this should be easy: $A=\pi r^2$. | ||
- | We are given current, and we can solve for electron current using the charge of an electron: $i = \frac{I}{q}$. | + | We are given current, and we can solve for electron current using the charge of an electron: $i = \frac{I}{|q|}$. |
We now have enough information to solve for the drift speed of electrons. We use positive numbers below, since we care only about speed for now, not direction. | We now have enough information to solve for the drift speed of electrons. We use positive numbers below, since we care only about speed for now, not direction. | ||
- | $$v_{avg} = \frac{I}{\pi r^2 n q}$$ | + | $$v_{avg} = \frac{I}{\pi r^2 n |q|}$$ |
Current ($I$), radius ($r$), electron density ($n$), and electron charge ($q$) are all things we know for our two wires. When we plug in the numbers, we get the following: | Current ($I$), radius ($r$), electron density ($n$), and electron charge ($q$) are all things we know for our two wires. When we plug in the numbers, we get the following: | ||
\begin{align*} | \begin{align*} | ||
- | v_{\text{avg, | + | v_{\text{avg, |
\end{align*} | \end{align*} | ||
+ | |||
+ | Notice that this is actually really slow! Depending on the material, the electron only travels somewhere between 1 mm - 1 cm per second on average. |