A long straight wire and a circular loop carrying currents...

A long straight wire and a circular loop carrying currents $I_{1}$ and $I_{2}$ respectively are in the same plane as shown in figure. If the magnetic field at the centre of the loop is zero, then :
(a) $I_{1}=I_{2}$
(b) $I_{1}=2 I_{2}$
(c) $I_{1}=2 \pi J_{2}$
(d) $I_{2}=2 \pi I_{1}$

Key Concepts

Magnetic Field of a Circular Current Loop

This concept refers to the magnetic field produced at the centre (or any point) of a loop carrying an electric current. The field at the centre of a circular loop of radius R carrying a steady current is derived using the Biot–Savart law and is given by an expression proportional to the current divided by the radius. This result is fundamental in understanding how currents circulated in closed loops generate magnetic fields in space.

Magnetic Field of a Straight Current?Carrying Conductor

A long, current?carrying wire produces a magnetic field in the space around it. For an infinitely long straight conductor, the magnetic field at a perpendicular distance from the wire is determined by Ampère’s law or the Biot–Savart law and decreases inversely with distance from the wire. This principle is critical for analyzing and comparing the contributions of different current elements to the resultant magnetic field.

Superposition Principle in Magnetism

The superposition principle states that when multiple sources of magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields produced by each source. This principle allows us to calculate conditions under which the magnetic fields from different current-carrying elements, such as a loop and a straight wire, exactly cancel each other, leading to a zero net field at a specific location.

Biot–Savart Law

The Biot–Savart law is a fundamental principle used to compute the magnetic field produced by a small element of current. It forms the basis for deriving the magnetic fields of both simple geometries such as straight wires and circular loops. This law is essential for understanding the spatial distribution of magnetic fields generated by currents in various configurations.

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