A current path shaped as shown in figure produces a mag....

A current path shaped as shown in figure produces a mag. field at point " $\mathrm{P}^{\prime \prime}$, the centre at the arc $\mathrm{BC}$. If the arc subtends an angle of $30^{\circ}$ and the radius of the arc is $0.6$ meter. What is the magnitude of the field at point $P$ if the current is 3 AMP? (a) $2.62 \times 10^{-6} \mathrm{~T}$ (b) $2.62 \times 10^{-7} \mathrm{~T}$ (c) $3.62 \times 10^{-7} \mathrm{~T}$ (d) $2.62 \times 10^{-8} \mathrm{~T}$

A current path shaped as shown in figure produces a mag. field at point " $\mathrm{P}^{\prime \prime}$, the centre at the arc $\mathrm{BC}$. If the arc subtends an angle of $30^{\circ}$ and the radius of the arc is $0.6$ meter. What is the magnitude of the field at point $P$ if the current is 3 AMP?
(a) $2.62 \times 10^{-6} \mathrm{~T}$
(b) $2.62 \times 10^{-7} \mathrm{~T}$
(c) $3.62 \times 10^{-7} \mathrm{~T}$
(d) $2.62 \times 10^{-8} \mathrm{~T}$

Key Concepts

Magnetic Permeability Constant (??)

The constant ??, known as the magnetic permeability of free space, appears in the equations describing the magnetic field generated by currents. It quantifies the ability of the medium (in this case, free space) to support the formation of a magnetic field and acts as a proportionality factor in electromagnetic equations such as those derived from the Biot-Savart law.

Biot-Savart Law

The Biot-Savart law is a fundamental principle in electromagnetism that relates the magnetic field generated by an element of current to its geometry and distance from the point at which the field is being calculated. It provides the basis for calculating the cumulative magnetic field produced by a current distribution by integrating small contributions from each infinitesimal segment of the current-carrying path.

Magnetic Field Due to a Circular Arc

For a circular arc, the magnetic field at the center of curvature is obtained by integrating the Biot-Savart law along the arc. The resulting expression shows that the field is directly proportional to the current and the angle subtended by the arc (expressed in radians) and inversely proportional to the radius of the arc, reflecting how the geometry of the current-carrying path influences the field strength.

Role of the Central Angle

The central angle of the arc determines the fraction of a complete circular loop represented by the arc. Since the magnetic field produced by a full circular loop is well defined, the field for an arc is calculated by multiplying the full-loop field by the ratio of the arc’s angle to 2?. This highlights the significance of angular span in magnetic field calculations for curved current paths.

Please give Ace some feedback