Magnetic Field Derivation for a Circular Coil
📌 The magnetic field (dB) produced by a small current element (dL) on a circular coil is calculated using the Biot-Savart Law: $dB = \frac{\mu_0}{4\pi} \frac{I\,dL \sin\theta}{r^2}$.
📐 Due to the geometry where the slant height (AP) is perpendicular to the coil's circumference, $\sin\theta = \sin 90^\circ = 1$, simplifying the field contribution to $dB = \frac{\mu_0}{4\pi} \frac{I\,dL}{r^2}$.
🔄 When resolving the vector contributions (dB) from opposite elements (like at points A and B), the components perpendicular to the axis ($\text{DB} \sin\phi$) add up, while the components parallel to the axis ($\text{DB} \cos\phi$) cancel each other out.

Total Magnetic Field on the Axis
🔗 The total magnetic field ($B$) is obtained by integrating the contributing component along the loop: $B = \int \text{DB} \sin\phi$.
🧮 Substituting $\sin\phi = \frac{a}{R}$ (where $a$ is the radius and $R$ is the distance AP) and $\int dL = 2\pi a$ (the perimeter), the formula simplifies to $B = \frac{\mu_0 I a^2}{2 R^3}$.
📏 Expressing $R$ in terms of the axial distance $x$ and radius $a$ ($R = \sqrt{x^2 + a^2}$), the final expression for the magnetic field on the axis at distance $x$ is $B = \frac{\mu_0 I a^2}{2 (x^2 + a^2)^{3/2}}$.

Special Cases and Center Field
🔄 For a coil with $n$ turns, the magnetic field is multiplied by $n$: $B_n = \frac{\mu_0 n I a^2}{2 (x^2 + a^2)^{3/2}}$.
⚫ The magnetic field at the center of the coil is found by setting the axial distance $x=0$, resulting in the formula $B_{\text{center}} = \frac{\mu_0 I}{2a}$.

Key Points & Insights
➡️ The derivation hinges on the geometric relationship ensuring $\sin\theta = 1$ for the magnetic field contribution from each element.
➡️ Cancellation of perpendicular components ($\text{DB} \cos\phi$) is crucial for the net field lying purely along the axis of the coil.
➡️ The magnetic field strength at the center of the coil ($B_{\text{center}} = \frac{\mu_0 I}{2a}$) represents the maximum field strength generated by the loop.

📸 Video summarized with SummaryTube.com on Nov 20, 2025, 04:55 UTC