Magnetic Fields and Biot-Savart Law
📌 The session covers defining magnetic fields, understanding the Biot-Savart Law and Ampère's Law, and applying them to calculate magnetic fields from electric currents.
🧲 The interaction between two parallel wires carrying currents ($I_1$ and $I_2$) demonstrates the existence of a magnetic field: parallel currents in the same direction attract, while opposite currents repel.
💡 The magnetic field intensity ($\mathbf{H}$) generated by a current-carrying wire creates a force (Lorentz force, $\mathbf{F} = I\mathbf{L} \times \mathbf{B}$) on a second wire; this field direction is determined using the right-hand rule (thumb along current, curled fingers indicate $\mathbf{H}$).
🔗 The magnetic flux density ($\mathbf{B}$) is related to the magnetic field intensity ($\mathbf{H}$) by the equation $\mathbf{B} = \mu_0 \mathbf{H}$.

Biot-Savart Law Formulation
📌 The Biot-Savart Law provides a method to calculate the magnetic field intensity ($\mathbf{H}$) generated by a current element ($I d\mathbf{l}$):
$$d\mathbf{H} = \frac{I d\mathbf{l} \times \mathbf{a}_R}{4\pi R^2}$$
where $R$ is the distance from the current element to the observation point, and $\mathbf{a}_R$ is the unit vector pointing from $d\mathbf{l}$ to the point.
🔗 For a closed loop, the total magnetic field is found by taking the closed line integral of the Biot-Savart expression.
⚡ For current distributions over surfaces (surface current density $\mathbf{K}$) or volumes (volume current density $\mathbf{J}$), the law is adapted:
$$d\mathbf{H} = \frac{\mathbf{K} \times \mathbf{a}_R dS}{4\pi R^2} \quad \text{or} \quad d\mathbf{H} = \frac{\mathbf{J} \times \mathbf{a}_R dV}{4\pi R^2}$$

Application: Magnetic Field of an Infinite Straight Wire
📌 Calculating the magnetic field intensity ($\mathbf{H}$) for an infinitely long straight wire along the $z$-axis, observed at a distance $\rho$ in the $xy$-plane (using cylindrical coordinates), involves integrating the differential form of the Biot-Savart Law.
📐 The setup requires defining the position vector $\mathbf{R} = \rho \mathbf{a}_{\rho} - z' \mathbf{a}_{z}$, where $R = \sqrt{\rho^2 + z'^2}$, and the current element $d\mathbf{l} = dz' \mathbf{a}_{z}$.
✅ The integration, performed by substitution using $\tan\theta = z'/\rho$, yields the result for the magnetic field intensity $\mathbf{H}$:
$$\mathbf{H} = \frac{I}{2\pi \rho} \mathbf{a}_{\phi}$$
(Note: The lecture derived the result as $\mathbf{H} = \frac{I}{2\pi \rho} \mathbf{a}_{\psi}$ where $\mathbf{a}_{\psi}$ represents the azimuthal unit vector, $\mathbf{a}_{\phi}$).

Key Points & Insights
➡️ Magnetic forces between wires arise because moving charges (currents) generate a magnetic field ($\mathbf{H}$) in the surrounding space.
➡️ Mastering the right-hand rule is essential for correctly determining the direction of the magnetic field ($\mathbf{H}$) around a current path and the resulting Lorentz force ($\mathbf{F}$).
➡️ For finite wires, the magnetic field intensity $\mathbf{H}$ at a point can be found by integrating the Biot-Savart Law between the limits $Z_1$ and $Z_2$, resulting in $\mathbf{H} = \frac{I}{4\pi \rho} (\sin\alpha_2 - \sin\alpha_1) \mathbf{a}_{\phi}$.

📸 Video summarized with SummaryTube.com on Nov 26, 2025, 03:32 UTC