Final answer:
The magnetic field at a point on the axis of a circular current-carrying loop is derived using the Biot-Savart Law. Due to the symmetry, the direction of the magnetic field at that point is parallel to the axis and determined by the right-hand rule. The correct direction of the magnetic field at this point is therefore described by option (c), following the right-hand rule.
Step-by-step explanation:
When using the Biot-Savart Law to deduce the expression for the magnetic field (B) at a point on the axis of a circular current-carrying loop, we must consider both the magnitude and direction of the field. The Biot-Savart Law relates magnetic fields to the currents which are their sources. In the case of a circular loop of radius (r), the magnetic field at a point (x) along the central axis of the loop is determined by integrating the magnetic contributions from each differential segment of the current path.
According to the Biot-Savart Law, the magnetic field dB, produced by a segment of current dI at point P is directly proportional to the current I, the length of the segment dl, the sine of the angle between dl and the line from dI to P, and is inversely proportional to the square of the distance between the segment and the point P. Mathematically, dB is a vector that lies in the plane perpendicular to the line joining dI and P. When applied to a circular loop, symmetry dictates that each differential segment creates a magnetic field component dB that points either directly towards or away from point P along the axis of the loop.
The direction of the magnetic field at point P can be determined by employing the right-hand rule. If you position your right thumb in the direction of the current, your fingers will curl in the direction of the magnetic field lines. This indicates that, for a point on the axis of a loop, the magnetic field's direction is parallel to the axis of the loop and follows the right-hand rule. Therefore, the correct option for the direction of the magnetic field in this scenario is (c) It follows the right-hand rule.