Detailed Summary
In Section 4.5, the magnetic field generated at a point along the axis of a circular current loop is evaluated. The circular loop, positioned in the XY-plane with its center at the origin (O) and radius R, carries a steady current (I). To determine the magnetic field (B) at a point located along the axis at a distance x from the center, we use the Biot-Savart law. The magnetic field dB due to an infinitesimal current element (Idl) is described by the formula:
$$
dB = \frac{\mu_0 I \, dl \times r}{4 \pi r^3}
$$
Where r is the displacement vector from dl to the axial point P, with its magnitude depending on both x and R. Integrating these contributions allows us to calculate the resultant magnetic field along the axis, revealing how the geometry of the loop shapes the magnetic field's distribution. The resulting formula for the magnetic field at any point on the axis and specifically at the center of the loop are provided:
$$
B = \frac{\mu_0 I R^2}{2 \left( x^2 + R^2 \right)^{3/2}}, \
B \text{ at the center: } B = \frac{\mu_0 I}{2R}
$$
These equations highlight not just the relationship between current, distance, and the resultant magnetic field, but also the underlying symmetry in the field created by current loops. The section also introduces the right-hand thumb rule for determining the direction of the magnetic field, thereby connecting the theoretical calculations with practical applications.