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+ | Sections 17.5 and 18.2 in Matter and Interactions (4th edition) | ||
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+ | / | ||
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+ | [[184_notes: | ||
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===== Current in Wires ===== | ===== Current in Wires ===== | ||
+ | In the last few pages of notes, we established that when connected to a battery there are surface charges in the wire that [[184_notes: | ||
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+ | {{youtube> | ||
+ | ==== Current in Different Parts of the Wire ==== | ||
+ | Given what you know about the electric field in the wire, how would you expect the electron current to compare in different parts of the wire? If the electric field is constant along the wire, each electron would feel a constant force along the wire. For every electron that leaves the negative plate of the battery, there is one returning to the positive plate of the battery. Thus, **at every point along the wire, the electron current is the same**. | ||
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+ | What if you added a light bulb to the circuit, how would you expect the electron current to compare? Do the electrons get "used up" in the light bulb? It turns out that electrons transfer electric energy into heat and light at the light bulb (we will talk about this more [[184_notes: | ||
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+ | //__In steady state__// we can rewrite the conservation of charge in terms of the electron current, called the " | ||
+ | $$Q_{in}=Q_{out}$$ | ||
+ | $$\frac{Q_{in}}{s}=\frac{Q_{out}}{s}$$ | ||
+ | $$i_{in}=i_{out}$$ | ||
+ | **This means for any given point in a circuit, the electron current entering that point or node must equal the electron current leaving that node**. | ||
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+ | ==== Why do we need an electric field? ==== | ||
+ | We have already established that there is an [[184_notes: | ||
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+ | ==== Drift speed in wire ==== | ||
+ | [{{ 184_notes: | ||
+ | Modeling all of these interactions for every electron in the electron current is quite complicated (or almost impossible). While there are several ways to model the electrons in the wire, we will use a model called the [[https:// | ||
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+ | Using the Drude Model, we can find the average drift velocity for the electrons in the wire. Starting with the momentum principle, we know | ||
+ | $$\frac{\Delta \vec{p}}{\Delta t}=\vec{F}_{net}$$ | ||
+ | which we could rearrange as: | ||
+ | $$\Delta \vec{p}=\vec{F}_{net}\Delta t$$ | ||
+ | where $\Delta \vec{p}$ is the change of momentum that occurs between collisions and $\Delta t$ is the time between collisions. In the wire, we know that the force on the electron comes from the electric field of the surface charges, so $\vec{F}_{net}=e\vec{E}$, | ||
+ | $$\Delta \vec{p}=e*\vec{E}*\Delta t$$ | ||
+ | If we //__assume that the electron loses all its momentum during each collision__//, | ||
+ | $$\Delta \vec{p}= \vec{p}-0= e*\vec{E}*\Delta t$$ | ||
+ | Since the speed of the electrons is much smaller than the speed of light, then: | ||
+ | $$\vec{v}=\frac{\vec{p}}{m_e}=\frac{e*\vec{E}*\Delta t}{m_e}$$ | ||
+ | However, the time between collisions is not the same for every electron (depending on each individual path) - sometimes that time is longer or shorter. If we take the average time between collisions, then that gives us the average drift velocity of the electrons in the wire: | ||
+ | $$\vec{v}_{avg}=\frac{e*\vec{E}*\Delta t_{avg}}{m_e}$$ | ||
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+ | Typically, we will define **electron mobility** as: | ||
+ | $$u=\frac{e*\Delta t_{avg}}{m_e}$$ | ||
+ | which is different for different metals and can be determined experimentally. So using electron mobility, we can write the drift velocity of the electrons as: | ||
+ | $$\vec{v}_{avg}=u\vec{E}$$. | ||
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+ | Likewise, we can combine this with the [[184_notes: | ||
+ | $$i=nAuE$$ | ||
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+ | ====Examples==== | ||
+ | * [[: | ||
+ | * Video Example: Drift Speed in Different Types of Wires | ||
+ | * [[: | ||
+ | * Video Example: Application of Node Rule | ||
+ | {{youtube> | ||
+ | {{youtube> |