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www.GOALias.blogspot.com4.5 MAGNETIC FIELD DUE TO A CURRENT ELEMENT,BIOT-SAVART LAWAll magnetic fields that we know are due to currents (or moving charges)and due to intrinsic magnetic moments of particles. Here, we shall studythe relation between current and the magnetic field it produces.It is given by the Biot-Savart’s law. Figure 4.9 shows a finiteconductor XY carrying current I. Consider an infinitesimalelement dl of the conductor. The magnetic field dB due to thiselement is to be determined at a point P which is at a distance rfrom it. Let θ be the angle between dl and the displacement vectorr. According to Biot-Savart’s law, the magnitude of the magneticfield dB is proportional to the current I, the element length |dl|,and inversely proportional to the square of the distance r. Itsdirection* is perpendicular to the plane containing dl and r .Thus, in vector notation,Idl× rdB∝r3µ Id ×4 r0 l r= π3[4.11(a)]where µ 0/4π is a constant of proportionality. The aboveexpression holds when the medium is vacuum.The magnitude of this field is,µ0I dlsinθdB=2[4.11(b)]4 π rwhere we have used the property of cross-product. Equation [4.11 (a)]constitutes our basic equation for the magnetic field. The proportionalityconstant in SI units has the exact value,µ0 −710 Tm/A=[4.11(c)]4πWe call µ 0the permeability of free space (or vacuum).The Biot-Savart law for the magnetic field has certain similarities aswell as differences with the Coulomb’s law for the electrostatic field. Someof these are:(i)Both are long range, since both depend inversely on the square ofdistance from the source to the point of interest. The principle ofsuperposition applies to both fields. [In this connection, note thatthe magnetic field is linear in the source I dl just as the electrostaticfield is linear in its source: the electric charge.](ii) The electrostatic field is produced by a scalar source, namely, theelectric charge. The magnetic field is produced by a vector sourceI dl.Moving Charges andMagnetismFIGURE 4.9 Illustration ofthe Biot-Savart law. Thecurrent element I dlproduces a field dB at adistance r. The ⊗ signindicates that thefield is perpendicularto the plane of thispage and directedinto it.* The sense of dl×r is also given by the Right Hand Screw rule : Look at the planecontaining vectors dl and r. Imagine moving from the first vector towards secondvector. If the movement is anticlockwise, the resultant is towards you. If it isclockwise, the resultant is away from you.143