4.9.2. Circular current loop as a magnetic dipole

In this section, we shall consider the elementary magnetic element: the current loop. We shall show that the magnetic field (at large distances) due to current in a circular current loop is very similar in behaviour to the electric field of an electric dipole. In Section 4.5, we have evaluated the magnetic field on the axis of a circular loop, of a radius R, carrying a steady current I. The magnitude of this field is

B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}

and its direction is along the axis and given by the right-hand thumb rule. Here, x is the distance along the axis from the centre of the loop. For x >> R, we may drop the R^2 term in the denominator. Thus,

B = \frac{\mu_0 I R^2}{2x^3}

Note that the area of the loop A = \pi R^2. Thus,

B = \frac{\mu_0 I A}{2\pi x^3}

This chunk explains how a circular current loop behaves like a small magnet, producing a magnetic field similar to that of an electric dipole. When a current flows through this loop, it generates a magnetic field that can be calculated based on its radius and the current's strength. At points far away from the loop (when the distance x is much greater than the radius R), the formula simplifies to show how the magnetic field diminishes with distance. Here, the area of the loop is also defined, which is crucial for calculating the magnetic moment.

Imagine a hula hoop that represents our current loop. When you give it a spin (representing the current), a wind circulates around it. If you move far away from the hula hoop, the wind (magnetic field) feels less intense, similar to how the magnetic field of the loop decreases with distance. This is akin to the way an electric dipole decreases in strength from a distance, like how the effect of a battery weakens as you move it away from a light bulb.

As earlier, we define the magnetic moment m to have a magnitude

m = I A.

Hence,

B \approx \frac{\mu_0 m}{4\pi x^3}.

In this chunk, we introduce the concept of the magnetic moment, which quantifies the strength and direction of a magnetic source. The magnetic moment for a current loop is calculated as the product of the current (I) and the area (A) of the loop. This definition plays an essential role in how we describe and analyze the behavior of magnetic fields produced by loops compared to electric charges.

Think of a spinning top. The faster it spins (analogous to a higher current), the more stable and prominent its spinning motion becomes (representing a stronger magnetic moment). A larger spinning top (representing a larger area) also shows a stronger spin compared to a small top. This analogy helps to understand how the magnetic moment encompasses both current and area: a larger loop with more current would create a stronger 'spin' in the magnetic field it produces.

The expression of Eq. [4.25(a)] is very similar to an expression obtained earlier for the electric field of a dipole. The similarity may be seen if we substitute,

µ → 1/ε₀.

Here, the chunk draws a comparison between electric dipoles and magnetic dipoles, suggesting that both behave similarly in terms of the fields they produce at distances. By replacing the magnetic properties with electric ones, we see how their fields display a compatible mathematical structure. This is significant because it indicates that the behavior of electric and magnetic fields may share underlying principles despite their fundamental differences.

Consider the way a pair of balloons can behave as dipoles when charged. They create an area of influence around them similar to electric and magnetic fields. Just as you can visualize electrical charges attracting or repelling, the way circulating currents in a magnetic dipole can be visualized shows that underlying these phenomena are similar concepts of field strengths, just manifesting in different physical forms.

Note, however, a fundamental difference: an electric dipole is built up of two elementary units — the charges (or electric monopoles). In magnetism, a magnetic dipole (or a current loop) is the most elementary element. The equivalent of electric charges, i.e., magnetic monopoles, are not known to exist.

This chunk highlights the essential distinction between electric and magnetic dipoles. While electric dipoles consist of two opposing charges (positive and negative), magnetic dipoles arise from loops of current which do not have individual magnetic 'monopole' charges. Instead, every magnetic field observed is a dipole field, spinning between two poles—not within one pole, which is unique to magnetism.

Imagine a pair of dice (representing electric charges) where one is red (positive) and the other blue (negative), showing how they work together in pairs as monopoles. In contrast, a magnetic dipole is like a bar magnet: you can't isolate one end of it without losing its magnetic property—every time you cut a magnet, you end up with two smaller magnets instead of a 'north' or 'south' monopole, illustrating the lack of independent magnetic charges.

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