Biot-Savart law

The Biot-Savart law is an equation describing the magnetic field generated by a current segment. Consider a point \small P in the vicinity of a wire carrying a current \small I (see diagram below).

The magnitude of \small I represents the amount of charges flowing through the wire (blue line). This implies that the amount of charges in an infinitesimal length of the wire \small d\vec{l} is proportional to \small Id\vec{l}. Since the magnetic field \small d\vec{B} at \small P is generated by the amount of moving charges in the wire, we have \small d\vec{B}\propto Id\vec{l}, or rather \small d\vec{B}\propto Id\vec{l}sin\theta because \small d\vec{B} is due to component of \small d\vec{l} that is perpendicular to \vec{R} (use right-hand rule to visualise).

The magnetic flux generated by \small I radiates spherically outward from every point along the infinitesimal length of the wire. For a particular \small I in a segment \small d\vec{l}, the amount of magnetic field lines radiated is a constant. As \small R increases, the surface area of the sphere becomes bigger. Therefore, the radiation passing through \small P must be proportional to the unit area of the sphere at \small R, i.e. \small d\vec{B}\propto \frac{1}{4\pi R^{2}}\hat{R}, where \small \hat{R} is the unit vector in the direction of \small \vec{R}. Finally, we would expect \small d\vec{B} to vary according to the material of the wire and the medium of the space between the wire and \small P. If we assume the wire to have negligible resistance and that the system is in a vacuum, we have

\small d\vec{B}=\frac{\mu_0Id\vec{l}sin\theta\hat{R}}{4\pi R^{2}}=\frac{\mu_0Id\vec{l}sin\theta\vec{R}}{4\pi R^{3}}=\frac{\mu_0Id\vec{l}\times\vec{R}}{4\pi R^{3}}\; \; \; \; \; \; \; \; 150

where \small \mu_0 is called the vacuum permeability or magnetic permeability of free space, and it represents the proportionality constant for the measurement of \small d\vec{B} at \small P in a vacuum.

The scalar form of eq150 is

\small d\vec{B}=\frac{\mu_0Idlsin\theta}{4\pi R^{2}}\; \; \; \; \; \; \; \; 151

The total field at \small P is

\small \vec{B}=\frac{\mu_0}{4\pi}\int \frac{Id\vec{l}\times\vec{R}}{R^{3}}\; \; \; \; \; \; \; \; 152

Eq150, eq151 and eq152 are different forms of the Biot-Savart law.

If the wire is a circle, the direction of \small d\vec{B} at \small P is determined by the right hand rule and is perpendicular to the plane formed by \small d\vec{l} and \small \vec{R}, where the angle \small \theta between \small d\vec{l} and \small \vec{R} is \small 90^{\circ} (see diagram below).

From eq151,

\small dB=\frac{\mu_0Idlsin90^{\circ}}{4\pi R^{2}}=\frac{\mu_0Idl}{4\pi R^{2}}\; \; \; \; \; \; \; \; 153

Let’s analyse the components of \small d\vec{B}. If we sum all the vectors of \small d\vec{B}_\perp at \small P from all current elements around the wire, they cancel out. The vectors of \small d\vec{B}_\parallel at \small P, however adds. Therefore, in scalar form, \small B=\int dB_\parallel, where \small dB_\parallel=dBcos\alpha, which when combined with eq153 gives

\small dB_\parallel=\frac{\mu_0Icos\alpha}{4\pi R^{2}}dl\; \; \; \; \; \; \; \; 154

Since \small R=\sqrt{r^{2}+z^{2}} and \small cos\alpha=\frac{r}{\sqrt{r^{2}+z^{2}}}, eq154 becomes \small dB_\parallel=\frac{\mu_0Ir}{4\pi(r^{2}+z^{2})^{3/2}}dl and

\small B=\int dB_\parallel=\frac{\mu_0Ir}{4\pi(r^{2}+z^{2})^{3/2}}\int dl

Substituting \small \int dl=2\pi r in the above equation

\small B=\frac{\mu_0Ir^{2}}{2(r^{2}+z^{2})^{3/2}}\; \; \; \; \; \; \; \; 155

For the case of \small B at point \small O, we have \small z, which is the length \small OP, equals to zero. So,

\small B=\frac{\mu_0I}{2r}\; \; \; \; \; \; \; \; 156

For the case of \small z\gg r, eq155 becomes \small B=\frac{\mu_0IA}{2\pi z^{3}}, where \small A=\pi r^{2}. Substitute eq60 and eq61 in \small B,

\small B=\frac{\mu_0\gamma}{2\pi z^{3}}L\; \; \; \; \; \; \; \; 157

Eq156 and eq157 are useful in constructing the orbit-orbit coupling term and the spin-orbit coupling term of the Hamiltonian.



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