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--- 184_notes:examples:week6_drift_speed [2017/09/26 15:51] – [Solution] tallpaul
+++ 184_notes:examples:week6_drift_speed [2021/06/08 00:49] (current) – schram45
@@ Line 1: / Line 1: @@
+[[184_notes:current|Return to current in wires]]
 =====Example: Drift Speed in Different Types of Wires=====
-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 of electrons in each wire? You may want to consult the table below.
-{{ 184_notes:6_n_table.jpg?800 |Properties of Metals}}
+[{{ 184_notes:6_n_table.jpg?700 |Properties of Metals}}]
 ===Facts===
@@ Line 8: / Line 10: @@
   * The zinc wire has $I=5 \text{ A}$, $r = 0.1 \text{ mm}$.
   * The charge of an electron is $q=-1.6\cdot 10^{-19} \text{ C}$.
+  * 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.
 ===Approximations & Assumptions===
-  * The table is accurate for our wires.
+  * The wires have circular cross-sections. This is typical of real wires and allows us to use the diameter of the wire to calculate the area properly.
-  * The wires have circular cross-sections.
+  * Using the [[184_notes:current|Drude model]] for electrons in the wire - the electrons are accelerated by the electric field, until they run into a positive nucleus, which reduces the speed back to zero.
-  * The wires do not experience any external electric field.
 ===Representations===
   * 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====
-We can look up electron
+We can use the [[184_notes:current|Drude model]] for electrons in the wire - the electrons are accelerated by the electric field, until they run into a positive nucleus, which reduces the speed back to zero. This is where our definition of drift speed comes from, so it is worth including it in our solution for reference.
+There are a lot of variables 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.
+</WRAP>
+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 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.
+$$v_{avg} = \frac{I}{\pi r^2 n |q|}$$
+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*}
+v_{\text{avg, Cu}} = 0.47 \text{ mm/s} &,& v_{\text{avg, Zn}} = 7.5 \text{ mm/s}
+\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.