Biot-Savart Law

Consider a closed electric circuit of general shape, fabricated from an idealized zero thickness wire, around which a current $I$ flows. According to Biot-Savart law, which is named after Jean-Baptiste Biot and Félix Savart, and which can be experimentally verified, the magnetic field generated by such a circuit is

$${\bf B} ({\bf r}) = \frac{\mu_0\,I}{4\pi} \oint \frac{d{\bf r}'\times ({\bf r} - {\bf r}')}{\vert{\bf r} - {\bf r}'\vert^3},$$ (2.226)

where $d{\bf r}'$ is an element of the wire, whose displacement is ${\bf r}'$, and the integral is taken around the whole circuit.

Figure 2.20: A Biot-Savart law calculation.

Consider an infinite straight wire, running along the $z$-axis, that carries a current $I$. See Figure 2.20. Let us reconstruct the magnetic field generated by the wire at point $P$ using the Biot-Savart law. Suppose that the perpendicular distance to the wire is $\rho$. It is easily seen that

$${\bf e}_z \times({\bf r} - {\bf r}') = \rho\,\,{\bf e}_\theta,$$ (2.227)

$$l = \rho\,\tan\phi,$$ (2.228)

$$dl = \frac{\rho}{\cos^2\phi} \,d\phi,$$ (2.229)

$$\vert{\bf r} - {\bf r}'\vert = \frac{\rho}{\cos\phi},$$ (2.230)

where $\theta$ is a cylindrical polar coordinate. (See Section A.23.) Hence,

$$d{\bf r}'\times({\bf r} - {\bf r}')= \frac{\rho^2\,{\bf e}_\theta}{\cos^2\phi} \,d\phi.$$ (2.231)

Thus, according to Equation (2.226), we have

$${\bf B} = \frac{\mu_0\,I}{4\pi} \int_{-\pi/2}^{\pi/2}\frac{\rho^2}{\cos^2\phi}\, \frac{1} {\rho^3 \,(\cos\phi)^{-3}} \,d\phi\,{\bf e}_\theta$$

$$=\frac{\mu_0 \,I}{4\pi \,\rho} \int_{-\pi/2}^{\pi/2} \cos\phi\,d\... ...mu_0 \,I}{4\pi \,\rho} \left[ \sin\phi\right]_{-\pi/2}^{\pi/2}\,{\bf e}_\theta,$$ (2.232)

which gives

$${\bf B} = \frac{\mu_0 \,I}{2\pi\, \rho}\,{\bf e}_\theta.$$ (2.233)

Thus, we conclude that the Biot-Savart law is a more general form of the familiar result (2.206) that is not restricted to long straight wires.

Consider a circular wire loop of radius $a$ that carries a current $I$. Suppose that the loop lies in the $x$-$y$ plane, and is centered on the origin. Let us use the Biot-Savart law to calculate the magnetic field generated by the coil along a perpendicular axis that passes through its center (i.e., along the $z$-axis). Let $z$ be the distance of the point of observation from the center of the loop, and let the angle $\theta$ parameterize position on the loop. Thus, we have

$${\bf r} =(0,~0,~z),$$ (2.234)

$${\bf r}' = (a\,\cos\theta,~a\,\sin\theta,~0),$$ (2.235)

where the right-hand sides of the previous two equations are Cartesian components. It follows that

$${\bf r}- {\bf r}' = (-a\,\cos\theta,~-a\,\sin\theta,~z),$$ (2.236)

$$\vert{\bf r} - {\bf r}'\vert = (a^2+z^2)^{1/2},$$ (2.237)

$$d{\bf r}' = (-a\,\sin\theta\,d\theta,~a\,\cos\theta\,d\theta,~0),$$ (2.238)

$$d{\bf r}'\times ({\bf r}-{\bf r}') = (a\,z\,\cos\theta\,d\theta,~a\,z\,\sin\theta\,d\theta,~a^2\,d\theta).$$ (2.239)

Thus, the Biot-Savart law, (2.226), yields

$$B_x = \frac{\mu_0\,I}{4\pi}\oint\frac{a\,z\,\cos\theta\,d\theta}{(a^2+z^2)^{3/2}}=0,$$ (2.240)

$$B_y = \frac{\mu_0\,I}{4\pi}\oint\frac{a\,z\,\sin\theta\,d\theta}{(a^2+z^2)^{3/2}}=0,$$ (2.241)

$$B_z = \frac{\mu_0\,I}{4\pi}\oint\frac{a^2\,d\theta}{(a^2+z^2)^{3/2}}=\frac{\mu_0\,I}{2}\,\frac{a^2}{(a^2+z^2)^{3/2}}.$$ (2.242)

Thus, the magnetic field generated on the $z$-axis is

$${\bf B} = \frac{\mu_0\,I}{2}\,\frac{a^2}{(a^2+z^2)^{3/2}}\, {\bf e}_z.$$ (2.243)

Suppose that we have two identical current loops of radius $a$. Let both loops be centered on the $z$-axis, and let the first lie in the plane $z= d$, and the second in the plane $z=-d$. Furthermore, suppose that a current $I$ flows around each loop in the same direction. By the principle of superposition, making use of the previous equation, the magnetic field generated on the $z$-axis by the two loops is

$$B_z = \frac{\mu_0\,I}{2}\left(\frac{a^2}{[a^2+(z-d)^2]^{3/2}} + \frac{a^2}{[a^2+(z+d)^2]^{3/2}}\right).$$ (2.244)

If we Taylor expand the previous expression about $z=0$ then we obtain

$$B_z= \frac{\mu_0\,I}{2}\,\frac{a^2}{(a^2+d^2)^{3/2}}\left\{2 + 3\left[\frac{(2\,d)^2-a^2}{(a^2+d^2)^2}\right]z^2+{\cal O}(z^4)\right\}.$$ (2.245)

Suppose that we wish to make the magnetic field in the region between the loops as uniform as possible. We can clearly achieve this goal if we adjust the spacing $2\,d$ between the loops in such a manner that the coefficient of $z^2$ in the previous expression is set to zero. In this case, the leading order non-constant term in the expansion is ${\cal O}(z^4)$. It can be seen that we need $2\,d=a$. In other words, the spacing between the loops must equal the radius of the loops. The approximately uniform magnetic field between the loops becomes

$$B_z = \left(\frac{4}{5}\right)^{3/2}\,\frac{\mu_0\,I}{a}.$$ (2.246)

A pair of current loops set up in this manner are known as Helmholtz coils.

Finally, we can generalize the Biot-Savart law, (2.226), to determine the magnetic field generated by a distributed current of density ${\bf j}({\bf r})$ by making the identification

$$I \,d{\bf r}={\bf j}({\bf r})\,dV.$$ (2.247)

Thus, we obtain

$${\bf B}({\bf r}) = \frac{\mu_0}{4\pi}\int \frac{{\bf j}({\bf r}')\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^3}\,dV',$$ (2.248)

where the volume integral is taken over all space.