(a) The magnetic field at O is only due to the curved path,...

(a) The magnetic field at $O$ is only due to the curved path, as for the line element, $d \vec{l} \uparrow \vec{r} .$ $$ \text { Hence, } \quad \vec{B}=\frac{\mu_{0} i}{4 \pi R} \pi(-\vec{k})=\frac{\mu_{0} i}{4 R}(-\vec{k}) $$ Thus $\vec{F}_{u}=i B(-\vec{j})=\frac{\mu_{0} i^{2}}{4 R}(-\vec{j})$ So, $F_{u}=\frac{\pi_{0} i^{2}}{4 R}=0 \cdot 20 \mathrm{~N} / \mathrm{m}$ (b) In this part, magnetic induction $\vec{B}$ at $O$ will be effective only due to the two semi infinite segments of wire. Hence $$ \begin{aligned} \vec{B} &=2 \cdot \frac{\mu_{0} i}{4 \pi\left(\frac{l}{2}\right)} \sin \frac{\pi}{2}(-\vec{k}) \\ &=\frac{\mu_{0} i}{\pi l}(-\vec{k}) \end{aligned} $$ Thus force per unit length, $$ \overrightarrow{F_{u}}=\frac{\mu_{0} l^{2}}{\pi l}(-\vec{i}) $$

(a) The magnetic field at $O$ is only due to the curved path, as for the line element, $d \vec{l} \uparrow \vec{r} .$
$$
\text { Hence, } \quad \vec{B}=\frac{\mu_{0} i}{4 \pi R} \pi(-\vec{k})=\frac{\mu_{0} i}{4 R}(-\vec{k})
$$
Thus $\vec{F}_{u}=i B(-\vec{j})=\frac{\mu_{0} i^{2}}{4 R}(-\vec{j})$
So, $F_{u}=\frac{\pi_{0} i^{2}}{4 R}=0 \cdot 20 \mathrm{~N} / \mathrm{m}$
(b) In this part, magnetic induction $\vec{B}$ at $O$ will be effective only due to the two semi infinite segments of wire. Hence
$$
\begin{aligned}
\vec{B} &=2 \cdot \frac{\mu_{0} i}{4 \pi\left(\frac{l}{2}\right)} \sin \frac{\pi}{2}(-\vec{k}) \\
&=\frac{\mu_{0} i}{\pi l}(-\vec{k})
\end{aligned}
$$
Thus force per unit length,
$$
\overrightarrow{F_{u}}=\frac{\mu_{0} l^{2}}{\pi l}(-\vec{i})
$$

Key Concepts

Biot-Savart Law

This law provides a method for calculating the magnetic field produced by a small segment of current-carrying conductor. It expresses the field as a vector quantity that depends on the current element, the distance from the element to the field point, and the sine of the angle between them. The law is integral to understanding how different segments of a current path contribute to the overall magnetic field at a given point.

Magnetic Field Due to Curved and Straight Wire Segments

The calculation of the magnetic field contribution from different segments of a conductor (curved and straight) relies on evaluating the effect of their geometrical arrangement using the Biot-Savart Law. For curved segments like arcs, the integration often yields factors involving the subtended angle, while for straight segments such as semi-infinite wires, geometric considerations lead to expressions involving sine of the angle between the wire and the observation point.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force explains how a current-carrying conductor in a magnetic field experiences a force. This force is given by the cross product of the current element and the magnetic field, implying that both the magnitude and the direction of the magnetic field have a direct impact on the force experienced by the conductor. The concept is essential for understanding the interaction between currents and magnetic fields in various configurations.

Superposition Principle in Magnetic Fields

When multiple segments of current-carrying conductors contribute to the magnetic field at a point, their individual fields must be vectorially added to obtain the net field. This principle—based on the linearity of Maxwell’s equations—allows for the magnetic effects of each segment to be calculated separately and then combined to understand the overall behavior of the system.

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