B IOT -S AVART L AW

B IOT -S AVART L AW All magnetic fields that we know are due to currents (or moving charges) and due to intrinsic magnetic moments of particles. Here, we shall study the relation between current and the magnetic field it produces. It is given by the Biot-Savart’s law. Figure 4.9 shows a finite conductor XY carrying current I . Consider an infinitesimal element d l of the conductor. The magnetic field d B due to this element is to be determined at a point P which is at a distance r from it. Let θ be the angle between d l and the displacement vector r . According to Biot-Savart’s law, the magnitude of the magnetic field d B is proportional to the current I , the element length |d l| , and inversely proportional to the square of the distance r . Its direction * is perpendicular to the plane containing d l and r . Thus, in vector notation, 3 I d ∝ l B 3 I d µ l [4.11(a)] where µ 0 /4 π is a constant of proportionality. The above expression holds when the medium is vacuum. The magnitude of this field is, d sin I l µ θ B [4.11(b)] where we have used the property of cross-product. Equation [4.11 (a)] constitutes our basic equation for the magnetic field. The proportionality constant in SI units has the exact value, 7 10 Tm/A µ [4.11(c)] We call µ 0 the permeability of free space (or vacuum). The Biot-Savart law for the magnetic field has certain similarities as well as differences with the Coulomb’s law for the electrostatic field. Some of these are: (i) Both are long range, since both depend inversely on the square of distance from the source to the point of interest. The principle of superposition applies to both fields. [In this connection, note that the magnetic field is linear in the source I d l just as the electrostatic field is linear in its source: the electric charge.] (ii) The electrostatic field is produced by a scalar source, namely, the electric charge.