A Current-Carrying Coil as a Magnetic Dipole — Question 2

Question 2

A zero axial component of a dipole field does not mean a zero magnetic field

A small coil at the origin has moment $\mu\hat z$. At distances large compared with its size, use $\vec B=\mu_0[3(\vec\mu\cdot\hat r)\hat r-\vec\mu]/(4\pi r^3)$. The observation point has polar angle theta from +z; define $C=\mu_0\mu/(4\pi r^3)$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the axial component Bz and cylindrical radial component Brho.

2. In the upper half-space, find the angle at which Bz vanishes and the total field there.

Question 2 - Solution

A zero axial component of a dipole field does not mean a zero magnetic field

(a) Method and calculation

$$\hat r=\sin\theta\hat\rho+\cos\theta\hat z,\quad\vec\mu\cdot\hat r=\mu\cos\theta.$$$$\boxed{B_z=C(3\cos^2\theta-1),\quad B_\rho=3C\sin\theta\cos\theta}.$$

(b) Interpretation and further calculation

$$B_z=0\Rightarrow\boxed{\theta=\cos^{-1}(1/\sqrt3)=54.7356^\circ}.$$$$B_\rho=3C\sqrt{2/3}/\sqrt3=\boxed{\sqrt2C},\quad\vec B=\sqrt2C\hat\rho\ne0.$$

Physical check

A component null depends on the measurement axis. The dipole field magnitude is nonzero at every finite radius away from the source in this ideal point-dipole model.