Magnetic field due to a circular loop

Picture a single circular loop of wire carrying a steady current $I$. Cut the loop into many tiny straight-ish segments. Each segment behaves like a short piece of straight wire, and by the right-hand thumb rule for a straight wire, each segment produces its own small circular field pattern curling around it.

Near the wire itself. Close to any point on the loop, the field looks much like the field of a straight wire at that same distance — nearly circular loops of field curling around the local direction of current flow.

At the centre. This is where the loop's shape matters. Every segment of the loop is roughly the same distance from the centre, and — crucially — the direction of the small field each segment produces at the centre is the same for every segment (perpendicular to the plane of the loop). So instead of partially cancelling, as fields from different parts of a complicated wire shape often do, all these small contributions add constructively. The result is a single strong, well-defined field pointing straight through the centre of the loop.

The overall pattern. Field lines emerge from one face of the loop, curve around outside the loop, and re-enter through the other face — exactly the pattern of a bar magnet's field, or of a flat disc magnet. One face behaves as an effective north pole (field lines leave it) and the other as an effective south pole (field lines enter it).

Direction — the right-hand thumb rule for a loop. Curl the fingers of your right hand in the sense the current flows around the loop; your thumb then points along the field direction at the centre. Reverse the current and the field at the centre reverses too.

Class 10 does not derive a formula for the central field, but the reasoning behind each dependence is straightforward once you see the loop as many small contributions adding up at one point.

More current, stronger field. Every small segment's contribution to the field grows with the current flowing through it (exactly as for a single straight wire). Since all the segments' contributions still add constructively at the centre, doubling the current in the loop doubles every one of those small contributions, so the total field at the centre also grows — the field at the centre increases as the current increases.

More turns, stronger field. A coil of $N$ turns wound tightly together, all carrying current in the same sense, is really $N$ loops stacked almost on top of each other. Each turn independently produces a field at the common centre, and because all turns carry current the same way round, their fields point in the same direction and simply add. So the field at the centre increases as the number of turns increases — this is precisely why real electromagnets use tightly wound coils rather than a single loop.

Smaller radius, stronger field. For a fixed current, shrinking the loop's radius brings every segment of wire physically closer to the centre. Just as a straight wire's field is stronger at points nearer to it, each segment's contribution at the centre grows as that segment moves closer — so a tighter loop concentrates the same current into a stronger central field. The field at the centre increases as the radius decreases. All three effects can act together or work against each other, which is exactly what the worked example below tests.

Question: A single circular loop carrying current $I$ produces a certain magnetic field at its centre. The same length of wire is now rewound into a tighter coil of 5 turns, each of correspondingly smaller radius, all turns carrying the same current $I$ in the same sense. Explain, factor by factor, how the field at the centre of the new coil compares with the field at the centre of the original single loop.

Solution:

Step 1. Identify what has changed. Two factors are different at once: (a) the number of turns has gone from 1 to 5, and (b) because the same total wire length is now wound into 5 smaller circles instead of 1 large one, each turn's radius has shrunk.

Step 2. Apply "more turns, stronger field" in isolation. With 5 turns carrying current in the same sense, all 5 turns' fields add at the shared centre — this factor alone would make the central field roughly 5 times stronger than a single turn of the same radius would give.

Step 3. Apply "smaller radius, stronger field" in isolation. Each of the 5 turns now has a smaller radius than the original single loop, and a smaller radius further strengthens the field each turn contributes at the centre.

Step 4. Combine the two effects. Both factors push the field in the same direction — up — so they reinforce rather than cancel. The coil's field at the centre is therefore markedly stronger than the field of the original single loop, by more than the factor of 5 that turns alone would give, because the radius shrink adds extra strengthening on top.

The coil's central field is much stronger than the single loop's, because increasing turns and decreasing radius work together — exactly the principle used to build practical electromagnets from ordinary wire.

Competitive stretch

Question: Loops A and B have the same number of turns and the same current, but A has a larger radius than B. Separately, if instead A carries twice the current of B while both have equal radius and turns, which loop wins the comparison in each case?

Step 1. Radius case: apply "smaller radius, stronger field." B has the smaller radius, so B has the stronger central field.

Step 2. Current case: apply "more current, stronger field." A carries twice the current, so A has the stronger central field.

In the radius comparison, loop B wins; in the current comparison, loop A wins — each factor is judged independently, holding the others fixed.

• Assuming the field is uniform everywhere inside the loop, as inside a solenoid — it is strongest and most well-defined right at the centre; closer to the wire itself the pattern is more complicated.
• Adding turns and assuming the field always strengthens — it only adds up if every turn carries current in the same sense; an oppositely wound turn would partly cancel the others instead of adding.
• Trying to recall or quote $B = \dfrac{\mu_0 N I}{2r}$ on a Class 10 answer — this formula is Class 12 material; stick to qualitative trends only.
• Mixing up which factor is responsible for a change — always name the one factor that changed (current, turns, or radius) before deciding whether the field grows or shrinks.

Ask the tutor: If a problem changes current, turns, and radius all at once, ask the AI tutor to walk through each factor's effect separately before combining them.