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Get instant insights and key takeaways from this YouTube video by Hero of the derivations.
Derivation of Magnetic Field for a Finite Conductor
📌 The derivation starts by applying the Biot-Savart Law to a small segment $dL$ of a current-carrying conductor, yielding .
📐 Geometric relationships within the constructed triangle lead to the substitution and expressions for and .
🔬 Substituting these into the Biot-Savart law simplifies the differential magnetic field to .
🧮 Integrating this expression over the relevant angular limits ( and ) yields the final formula for the magnetic field $B$: .
Magnetic Field for an Infinite Conductor (Special Case)
♾️ For a conductor of infinite length, the angles and at the ends approach .
➕ Substituting and into the finite conductor formula results in .
💡 This derived equation, , represents the magnetic field produced by an infinitely long, straight, current-carrying wire at a perpendicular distance $a$.
Key Points & Insights
➡️ The Biot-Savart Law is the fundamental tool used to calculate the magnetic field contribution ($dB$) from infinitesimal current elements ($dL$).
➡️ Careful geometric substitution (using and ) converts the integral variable from length ($L$) to the angle ().
➡️ The angular limits must be taken relative to the perpendicular axis; for standard orientation, one angle will typically be positive and the other negative (or treated as positive and integrated from to ).
➡️ The final formula for a finite conductor is , where $a$ is the perpendicular distance.
📸 Video summarized with SummaryTube.com on Nov 20, 2025, 04:14 UTC
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Full video URL: youtube.com/watch?v=NyUUD9c8p2M
Duration: 17:36
Get instant insights and key takeaways from this YouTube video by Hero of the derivations.
Derivation of Magnetic Field for a Finite Conductor
📌 The derivation starts by applying the Biot-Savart Law to a small segment $dL$ of a current-carrying conductor, yielding .
📐 Geometric relationships within the constructed triangle lead to the substitution and expressions for and .
🔬 Substituting these into the Biot-Savart law simplifies the differential magnetic field to .
🧮 Integrating this expression over the relevant angular limits ( and ) yields the final formula for the magnetic field $B$: .
Magnetic Field for an Infinite Conductor (Special Case)
♾️ For a conductor of infinite length, the angles and at the ends approach .
➕ Substituting and into the finite conductor formula results in .
💡 This derived equation, , represents the magnetic field produced by an infinitely long, straight, current-carrying wire at a perpendicular distance $a$.
Key Points & Insights
➡️ The Biot-Savart Law is the fundamental tool used to calculate the magnetic field contribution ($dB$) from infinitesimal current elements ($dL$).
➡️ Careful geometric substitution (using and ) converts the integral variable from length ($L$) to the angle ().
➡️ The angular limits must be taken relative to the perpendicular axis; for standard orientation, one angle will typically be positive and the other negative (or treated as positive and integrated from to ).
➡️ The final formula for a finite conductor is , where $a$ is the perpendicular distance.
📸 Video summarized with SummaryTube.com on Nov 20, 2025, 04:14 UTC
Find relevant products on Amazon related to this video
Tool
Shop on Amazon
Fundamental Tool
Shop on Amazon
Best Tool
Shop on Amazon
Best Fundamental Tool
Shop on Amazon
As an Amazon Associate, we earn from qualifying purchases

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