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Coils and probe
Two identical coils carrying the same current in the same sense; the field is most uniform when their separation equals their radius.
Model checks
The science
What is modelled
Two thin circular current loops, each with N turns, on a common axis. The field is computed by numerically integrating the Biot-Savart law over each loop (a polygon of short segments), then compared against closed forms on the axis.
Equations
- Biot-Savart
dB = (μ0 N I / 4π) dl × r / r3. - On the axis
B(x) = (μ0 N I R2 / 2) [ (R2 + (x - d/2)2)-3/2 + (R2 + (x + d/2)2)-3/2 ]; for d = R the centre value is(4/5)3/2 μ0 N I / Rand the second derivative vanishes.
Honest limits
Ideal filamentary loops: no wire thickness, iron, heating or time dependence. A small regularising distance avoids a singularity on the wire itself. Field lines in the plane z = 0 are traced by RK4.
What to try
- Choose Too close and Too far apart: the field becomes dipped or peaked at the centre; at d = R the curve is flat.
- Move the probe off the axis and watch the field magnitude and direction change.
- Reverse the current to a negative value: every field line reverses, the magnitude stays the same.