Introduction
A general part of electrodynamics is concerned with the investigation of the magnetic fields produced by current-carrying conductors. The most relevant geometry is the circular current loop due to its natural symmetry and that it can be applied to real devices like solenoids, electromagnets, as well as the magnetic properties of atomic systems.
The passage of an electric current around a wire in a circular form creates a magnetic field that surrounds the space. In contrast to the situation with a straight conductor, where the field is concentrated within concentric circles, the circular loop provides a more complicated field distribution, which is still analytically solvable. The magnetic field component in the direction of the symmetry axis of the loop is of special interest because the axial symmetry allows obtaining short analytic expressions.
The derivation of the analytical formula of the magnetic field on the axis of a circular loop of current through the Biot-Savart law, the explanation of the field direction through the right-hand rule, and some salient examples and applications will then follow in this exposition.
The Setup: Current Loop and Axis
Imagine a circular loop of radius R, lying flat in the y-z plane with its center at the origin O. A steady current I flows through the loop, as shown in Fig. 1.
The axis of the loop lies along the x-direction, and we want to calculate the magnetic field at a point P on this axis, at a distance x from the center of the loop.
This setup has rotational symmetry about the axis of the loop. That symmetry will simplify the otherwise complicated vector sum of the magnetic field contributions from each element of the loop.
Biot–Savart Law and Elemental Contribution
The Biot–Savart law provides the magnetic field dB due to a current element I dl at a point in space:
dB = μ0/4π × I(dl × r)/r3
Here:
- dl = vector length element of the current
- r = displacement vector from the element to the observation point
- r = magnitude of r
- μ0 = permeability of free space
Now consider an element dl on the loop. Its displacement vector to the point P on the axis is r, with magnitude:
r = √(x2 + R2)
This is because the point P is at a distance x along the axis, and the element is at a radius R in the loop plane.
Also, dl is always perpendicular to the radius vector drawn in the plane of the loop. Thus,
|dl × r| = r dl
So the magnitude of the elemental magnetic field becomes:
dB=μ0/4π × Idlr/r3=μ0/4π × Idl/(x2+R2)
Net Magnetic Field on the Axis
The direction of dB is shown in Fig. 1. It is perpendicular to both dl and r. Therefore, each elemental field dB has two components:
- A component along the x-axis (dBx)
- A component perpendicular to the x-axis
Because of symmetry, the perpendicular components (in the y-z plane) cancel each other out when summed around the loop. Only the axial components survive.
Magnetic Field at the Center of the Loop
At the center of the loop, x=0. Substituting in the formula:
B0 = μ0I / 2R
This result is important because it shows that the magnetic field at the center of a circular loop is directly proportional to the current and inversely proportional to the radius.
For a loop of smaller radius with the same current, the field is stronger.
Direction of the Field: Right-Hand Thumb Rule
In order to determine the direction of the magnetic field, the right-hand thumb rule is used.
Turn the fingers of the right hand in the direction the loop goes on the left; the thumb will then point along the axis, hence showing the direction of the field.
Figure 2 demonstrates that the magnetic field lines are closed circles, the circle has its origin on one side of the circle, and it enters the circle on the other side. The loop is therefore acting like a little magnetic dipole: one side of the loop where the field is leaving is acting like a north pole, the other side, where the field is reentering, is a south pole.
Example Applications and Problems
Example 4.6: Semi-Circular Arc of Current
A straight conductor is reconfigured into a semicircular arc of radius equal to 2.0 cm with a current of 12 A, as shown in Fig. 3(a). Calculate the magnetic field in the centre of the arc.
- (a) In the straight segments, the angle between the infinitesimal element, dl, and the radius vector, r, is zero and therefore dl x r=0. These sections do not, thus, add anything to the magnetic field.
- (b) In the case of the semicircular arc, the sum of contributions of all the elements of the difference is constructive. With the right-hand rule, the resulting magnetic field is into the plane of the page. The diameter of it is half that of a full circular loop:
B = μ0I / 4R
For the given values, this gives approximately 1.9×10−4 T.
- (c) If the arc is bent in the opposite direction as in Fig. 3(b), the direction of B simply reverses.
Example 4.7: Tightly Wound Coil with Many Turns
Now consider a coil with N=100 turns, radius R=10 cm, carrying current I=1 A.
Since each turn produces the same field at the center, the net field is simply N times the single-turn result:
B = μ0NI / 2R
Substituting values:
B = 4π×10−7 × 100 × 1 / (2 × 0.1)
B ≈ 6.28×10−4 T
This example illustrates how multiple turns reinforce the field, a principle used in solenoids and electromagnets.
Visualizing Magnetic Field Lines
Magnetic field lines due to a circular loop, shown in Fig. 2, have a structure very similar to that of a bar magnet. At large distances (compared to the loop radius), the loop behaves like a magnetic dipole, with a dipole moment:
m = I ⋅ A = I πR2
Here, A is the area of the loop. This is why circular current loops are often treated as tiny magnetic dipoles in advanced physics.
Conclusion
The magnetic field on the axis of a circular current loop beautifully demonstrates how symmetry and the Biot–Savart law combine to simplify a complex vector problem. The result:
B = μ0IR2 / 2(x2 + R2)3/2
captures the essence of the system. At the center, it reduces to:
B0 = μ0I / 2R
This derivation also shows why loops and coils are powerful building blocks of electromagnets and why they are used in sensors, motors, and countless devices. The examples of semicircular arcs and tightly wound coils (Examples 4.6 and 4.7) further illustrate how variations of the loop geometry affect the magnetic field.
From the elegant lines of Fig. 2 to the practical setups in Fig. 3, the circular loop remains one of the most illuminating cases in the study of electromagnetism.