4.5. MAGNETIC FIELD ON THE AXIS OF A CIRCULAR CURRENT LOOP

In this section, we shall evaluate the magnetic field due to a circular coil along its axis. The evaluation entails summing up the effect of infinitesimal current elements (I dl) mentioned in the previous section. We assume that the current I is steady and that the evaluation is carried out in free space (i.e., vacuum).

This chunk introduces the aim of the section: to calculate the magnetic field generated by a circular coil when viewed from its axis. It signifies that, to find this magnetic field, we will consider many small segments of the coil, each carrying a current. The assumption is that the current is constant and we’re observing it in a vacuum where external factors do not affect the results.

Think of the circular coil like a group of small people standing in a circle, all of them holding hands and pulling with equal strength. This collective effort influences what happens in the center of this circle, much like how the small current segments influence the magnetic field at the center of the coil.

Fig. 4.9 depicts a circular loop carrying a steady current I. The loop is placed in the plane with its centre at the origin O and has a radius R. The x-axis is the axis of the loop. We wish to calculate the magnetic field at the point P on this axis. Let x be the distance of P from the centre O of the loop.

Here, the chunk discusses the geometry involved in the situation being analyzed. We have a circular loop with a radius R and a current I flowing through it. The center of the loop is at the origin of a coordinate system, and we are particularly interested in the magnetic field effect at point P, which lies along the x-axis at a distance x from the center. This setup is crucial for calculations later as it provides clear parameters for equations.

Imagine drawing a circle on a piece of paper. The center is like a busy city, and the area within the circle represents a neighborhood. As people in the neighborhoods move (like our current), they impact what happens in the city center. Similarly, each part of the loop contributes to the magnetic field at point P.

Consider a conducting element dl of the loop. The magnitude dB of the magnetic field due to dl is given by the Biot-Savart law [Eq. 4.7(a)].

dB = µ₀(I dl × r) / (4π r³)

where r² = x² + R². Further, any element of the loop will be perpendicular to the displacement vector from the element to the axial point.

In this section, we use the Biot-Savart law, which provides a way to calculate the magnetic field generated by a small segment of current. The formula states that the magnetic field contribution dB from a small segment dl of the loop at point P is dependent on the current (I), the small length of the wire (dl), and the distance (r) from the segment to point P. Note that r depends on both x and R, the radius of the loop. This setup allows us to calculate the total magnetic field by integrating all these small contributions.

Imagine each person (the current element) holding onto a stick (the dl) which pushes against a nearby object. The strength of the push (the dB) depends on how strong they are pulling (the current I) and how far they are from the object. To know the total impact on the object at the center (point P), we need to consider the contributions from all the people around the circle.

The direction of dB is shown in Fig. 4.9. It is perpendicular to the plane formed by dl and r. It has an x-component dBx and a component perpendicular to the x-axis, dB⊥. When the components perpendicular to the x-axis are summed over, they cancel out and we obtain a null result.

This part of the evaluation deals with the directionality of the magnetic field created by each current segment of the loop. According to the right-hand rule, the direction of the magnetic field dB is determined to be perpendicular to the plane created by the current segment dl and the line connecting the segment to point P (r). When analyzing contributions along the axis and those that are perpendicular, it becomes apparent that the perpendicular influences cancel each other out, leaving only the contribution along the axis of interest.

Consider rotating a fan in a room. The blades exert wind (like the magnetic field) in various directions. If you look closely, some parts of the wind might push against the walls, but those forces cancel each other out, while only the wind that moves directly forward (along the specified direction) matters to the person standing in front of the fan.

Thus, only the x-component survives. The net contribution along x-direction can be obtained by integrating dBx over the loop. The summation of elements dl over the loop yields 2πR, the circumference of the loop. Thus, the magnetic field at P due to entire circular loop is B = (μ₀I R²) / [2(x² + R²)³/2].

This final chunk sums it all together by integrating the x-component of the magnetic field over the entire loop. We approximate the contributions along the axis and sum the entire contribution into a single equation. From this, we find the total magnetic field expression at point P on the axis, creating a relationship based on the current, geometry of the loop, and distance from the center.

Picture collecting all the wind gusts from every blade of a fan into one large blast you would feel directly in front of it. You would take into account every tiny push from each blade to calculate the total push of air (the total magnetic field) felt at your face. Just as each gust combines to create one strong wind, each piece’s contribution combines to give us the total magnetic effect at point P.

--