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| 184_notes:examples:week9_detecting_b [2017/10/24 01:11] – dmcpadden | 184_notes:examples:week9_detecting_b [2021/07/05 21:58] (current) – [Solution] schram45 | ||
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| =====Magnetic Field near a Moving Charge===== | =====Magnetic Field near a Moving Charge===== | ||
| You are a collector of magnetic field detectors. A fellow detector collector is trying to trim down her collection, and so it's your job to see if an old detector is still working properly, in which case it's yours! Today, you are a magnetic field detector collector, inspector, and hopefully a selector. First, you run a test in which a charged particle ($q = 15 \text{ nC}$) is sent through the detector, and you look at the detector' | You are a collector of magnetic field detectors. A fellow detector collector is trying to trim down her collection, and so it's your job to see if an old detector is still working properly, in which case it's yours! Today, you are a magnetic field detector collector, inspector, and hopefully a selector. First, you run a test in which a charged particle ($q = 15 \text{ nC}$) is sent through the detector, and you look at the detector' | ||
| - | {{ 184_notes: | + | [{{ 184_notes: |
| ===Facts=== | ===Facts=== | ||
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| ===Approximations & Assumptions=== | ===Approximations & Assumptions=== | ||
| - | | + | * We are only interested in the $B$-field at this specific moment in time: As the particle moves some its parameters may change (i.e. velocity, charge...). This assumption gives use fixed variables to work with at this snapshot in time, simplifying down the complexity of the model. |
| - | | + | |
| ===Representations=== | ===Representations=== | ||
| * We represent the Biot-Savart Law for magnetic field from a moving point charge as | * We represent the Biot-Savart Law for magnetic field from a moving point charge as | ||
| $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | ||
| + | <WRAP TIP> | ||
| + | ===Approximation=== | ||
| + | We must approximate the particle as a point particle in order to use the magnetic field equation above. Since the problem doesn' | ||
| + | </ | ||
| * We represent the situation with diagram given above. | * We represent the situation with diagram given above. | ||
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| \vec{B}_3 &= \frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}_3}{{r_3}^3} = 4.2 \text{ nT } \hat{z} | \vec{B}_3 &= \frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}_3}{{r_3}^3} = 4.2 \text{ nT } \hat{z} | ||
| \end{align*} | \end{align*} | ||
| + | |||
| + | Observation location 3 is the furthest away from our moving point charge and we would expect it to have a smaller magnetic field than location 2, this is reflected in our solution. We also expected the magnetic field at location 1 to be 0 since the velocity and separation vector are parallel for this point (this is always a good thing to look for when approaching a problem with cross products). | ||