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 $dB = \frac{\mu_0}{4\pi} \frac{I \, dL \sin\theta}{r^2}$.
📐 Geometric relationships within the constructed triangle lead to the substitution $\sin\theta = \cos\phi$ and expressions for $dL = a \sec^2\phi \, d\phi$ and $r = a \sec\phi$.
🔬 Substituting these into the Biot-Savart law simplifies the differential magnetic field to $dB = \frac{\mu_0 I}{4\pi a} \cos\phi \, d\phi$.
🧮 Integrating this expression over the relevant angular limits ($\phi_1$ and $\phi_2$) yields the final formula for the magnetic field $B$: $B = \frac{\mu_0 I}{4\pi a} (\sin\phi_1 + \sin\phi_2)$.

Magnetic Field for an Infinite Conductor (Special Case)
♾️ For a conductor of infinite length, the angles $\phi_1$ and $\phi_2$ at the ends approach $90^\circ$.
➕ Substituting $\phi_1 = 90^\circ$ and $\phi_2 = 90^\circ$ into the finite conductor formula results in $B = \frac{\mu_0 I}{4\pi a} (1 + 1) = \frac{\mu_0 I}{2\pi a}$.
💡 This derived equation, $B_{\infty} = \frac{\mu_0 I}{2\pi a}$, 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 $\tan\phi$ and $\cos\phi$) converts the integral variable from length ($L$) to the angle ($\phi$).
➡️ 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 $-\phi_2$ to $\phi_1$).
➡️ The final formula for a finite conductor is $B = \frac{\mu_0 I}{4\pi a} (\sin\phi_1 + \sin\phi_2)$, where $a$ is the perpendicular distance.

📸 Video summarized with SummaryTube.com on Nov 20, 2025, 04:14 UTC