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### Pro Tips for Solving Mathematical Problems by (O.J.B.):
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| Tip | Explanation/ Details |
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| **Tip** | **Explanation/ Details** |
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| :--- | :--- |
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| Substitute giveb variables with custom <br> variables (e.g., AbcdEfG) | Solve the equation using your own variables, then mirror the <br> steps onto the original problem for proportional reasoning. |
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| Interpret the equal sign as <br> "converts to" | Thinking of "=" as "converts to" can facilitate substitutions & <br> manipulations in other mathematical expressions. |
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| *Spinors are two-component objects, they do return to the same quantum state after a 4π rotation but not after 2π.|
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| **Key terms**: irreducible representations, experimental consequences |
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### How to Generate a Pure Electric Field:
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| **Method** | **Mechanism** | **Key Considerations** | **Applications** |
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|-------------------------|-----------|--------------------|--------------|
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| **Photoelectric Effect** | Light ejects electrons from a surface, creating charge imbalance | Requires photon energy > work function; works best in vacuum | Spacecraft charging, photoemission devices |
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| **Charge Separation** | Electrons are emitted and collected on a secondary surface | Requires an electron collector; prevents neutralization | Photoemission-based capacitors, charge storage |
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| **Vacuum Conditions** | Electrons travel freely, leading to sustained electric fields | No surrounding medium to neutralize charge | Electron beam devices, vacuum tube applications |
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| **Voltage Bias Application** | A potential difference guides photoelectrons to a specific region | Ensures continuous charge separation | Controlled electron beams, energy harvesting |
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| **Solar Wind or UV Radiation** | Continuous electron emission from a surface in space | Happens naturally on airless bodies like the Moon | Spacecraft charging, lunar dust levitation |
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| **Photoemission Cathodes** | Use of a light-activated electron source in a circuit | Requires efficient cathode material | Photocathodes for electron guns, free-electron lasers |
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| **Exposure to UV Light** | UV photons ionize surfaces, causing electron ejection | Effective in high-energy UV environments like space or high-intensity lamps | Static charge buildup, atmospheric electricity effects, UV-sensitive detectors |
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### Definition of Arbitrary & Arbitrary Units:
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| **Term** | **Definition** |
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## All The Physics You Need, Curated by (O.J.B.):
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| Topic | Source |
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| **Topic** | **Source** |
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|---|---|
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| How to Succeed at Physics Without Really Trying | [Physics with Elliot](https://youtu.be/kC6U900CvwY) |
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| The Most Important Math Formula for Understanding Physics | [Physics with Elliot](https://youtu.be/HQsZG8Yxb7w) |
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## Additional Physics & Mathematics Resources:
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| Topic | Source |
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| **Topic** | **Source** |
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| Introduction to Mathematical Reasoning Playlist | [Knop's Course](https://youtube.com/playlist?list=PLxYr6TaF_SDU9ovvxPGAxI0dXfPgwhfBl&si=aTWKeeDbTgRDBDsn) |
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| General Mathematical Playlists | [Faculty of Khan](https://www.youtube.com/@FacultyofKhan/playlists) |

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