Physics 2212, Sim 09: Magnetic Fields
Eric Murray, Fall 2020
Required Advance Reading
From the Biot-Savart Law, we can find the magnetic field on the axis of a circular current loop. In terms of magnitudes,
B = (μ0 / 4π) (I Δs sinφ) ⧸ r2 ⇒ Bloop = (μ0 / 2) IR2 ⧸ (z2 + R2)3/2
where the loop of radius R is carrying current I and the field magnitude is determined at a distance z along the axis from the center of the loop. In some circumstances, this field magnitude may be approximated by
Bloop ≈ (μ0 / 4π) μ ⧸ z3
where μ is the magnetic dipole moment of the loop. You should make sure you know when this approximation is valid.
Let us gather the constants for a particular loop together and call them C, so
Bloop ≈ C ⧸ z3 ≈ Cz-3
Taking the natural log of both sides yields
ln(Bloop) ≈ -3 ln(z) + C'
where C' is a new constant. This means that a graph of ln(Bloop) as a function of ln(z) should be a line of slope -3. Note that if the field magnitude depended on some other power of z, that power could be determined by this analysis, as it would be the slope of the graph.
You will find the power relationship between a simulated current loop and distance, as well as that between a simulated permanent magnet and distance. (Although a permanent magnet is not a current loop, it has a dipole moment, so the above analysis is the same from the point μ was introduced.)