A wire P Q R S shown in the figure carries a current i. The...

A wire $P Q R S$ shown in the figure carries a current $i$. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular part of the wire is given by (a) $\frac{\mu_{o} i}{2 r}$ (b) $\frac{\mu_{o} i}{8 r}$ (c) $\frac{3 \mu_{e} i}{4 r}$ (d) $\frac{3 \mu_{o} i}{8 r}$

A wire $P Q R S$ shown in the figure carries a current $i$. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular part of the wire is given by
(a) $\frac{\mu_{o} i}{2 r}$
(b) $\frac{\mu_{o} i}{8 r}$
(c) $\frac{3 \mu_{e} i}{4 r}$
(d) $\frac{3 \mu_{o} i}{8 r}$

Key Concepts

Superposition Principle in Magnetism

The superposition principle states that the total magnetic field produced by multiple current-carrying segments or configurations is the vector sum of the magnetic fields produced by each individual segment. This principle allows complex configurations, like a wire combining circular arcs and straight segments, to be analyzed by separately calculating the magnetic field due to each component and then summing the results to obtain the net field at a specific point.

Magnetic Field Due to a Finite Straight Conductor

The magnetic field generated by a finite straight conductor carrying current is not uniform and depends on the positions of the endpoints relative to the point of interest. It can be calculated using a specific formula derived from the Biot-Savart Law, which involves the sine of the angles between the line joining the ends of the conductor to the point and the conductor itself. This concept is useful when analyzing contributions to the magnetic field from straight segments adjacent to curved parts in a composite wire structure.

Magnetic Field Due to a Circular Arc

A circular arc carrying current produces a magnetic field at the center of the circle that can be calculated as a fraction of the field produced by a complete circular loop. Specifically, the magnetic field produced by an arc is proportional to the angle subtended by the arc at the center, relative to the 360° (or 2? radians) of a full circle. This concept is critical when dealing with partial loops in current carrying conductors.

Biot-Savart Law

The Biot?Savart Law is a fundamental principle in electromagnetism that allows the calculation of the magnetic field produced by a steady current. It states that the magnetic field at a given point in space is the integral of contributions from each infinitesimal segment of current, with each contribution inversely proportional to the distance from the segment and dependent on the sine of the angle between the segment and the line connecting it to the point. This law is essential for determining the magnetic fields of complex current geometries such as arcs and straight wires.

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