Calculating Magnetic Field for a Straight Wire using Biot-Savart Law
📌 The procedure involves using the Biot-Savart Law to calculate the magnetic field ($\text{B}$) produced by a straight wire carrying current ($I$).
📐 Key steps include considering a small current element ($\text{dl}$) and defining the position vector ($\text{r}$) and the angle ($\theta$) between them; $\text{B}$ is calculated via integration.
🔄 The variables ($\text{dl}$, $\text{r}$, $\theta$) must be converted into a single variable (e.g., $\phi$, the angle from the perpendicular line) to perform the integration over the wire's length ($L$).

Integration and Final Formula Derivation
🔍 The relationship $\text{dl} = \text{r} d\phi / \cos \phi$ and $\sin \theta = \cos \phi$ are used to transform the integral expression from terms of $\text{dl}$ and $\theta$ to terms of $\phi$.
➗ The integration of $\cos \phi$ yields $\sin \phi$. The magnetic field ($\text{B}$) for a straight wire of finite length is found by integrating from limits $\phi_1$ to $\phi_2$:
$$\text{B} = \frac{\mu_0 I}{4 \pi a} (\sin \phi_2 - \sin \phi_1)$$
where $a$ is the perpendicular distance from the wire to the point.

Special Cases and Boundary Conditions
⚫ If the point is directly opposite one end of the wire (e.g., $\phi_1 = 0^\circ$) and the other end is at $\phi_2 = 90^\circ$, the field is $\text{B} = \frac{\mu_0 I}{4 \pi a}$.
♾️ For an infinitely long straight wire, $\phi_1 \rightarrow -90^\circ$ and $\phi_2 \rightarrow 90^\circ$, resulting in the simplified formula $\text{B} = \frac{\mu_0 I}{2 \pi a}$.

Key Points & Insights
➡️ The Biot-Savart law requires converting all spatial variables ($\text{dl}$, $\text{r}$, $\theta$) into a single angular variable ($\phi$) relative to the integration point before performing integration.
➡️ Be mindful of the sign convention when setting integration limits ($\phi_1$ and $\phi_2$); angles below the perpendicular are typically taken as negative.
➡️ Special case results, such as the field for an infinite wire ($\text{B} = \frac{\mu_0 I}{2 \pi a}$), are derived by setting the limits of integration to $-90^\circ$ and $90^\circ$.

📸 Video summarized with SummaryTube.com on Nov 20, 2025, 04:02 UTC