Section

[ . (a)] where I is the total current through the surface. The integral is taken over the closed loop coinciding with the boundary C of the surface. The relation above involves a sign-convention, given by the right-hand rule.

Let the fingers of the right-hand be curled in the sense the boundary is traversed in the loop integral “B.dl. Then the direction of the thumb gives the sense in which the current I is regarded as positive. For several applications, a much simplified version of Eq. [ .

(a)] proves sufficient. We shall assume that, in such cases, it is possible to choose the loop (called an amperian loop) such that at each point of the loop, either FIGURE . EXAMPLE . (i) B is tangential to the loop and is a non-zero constant B, or (ii) B is normal to the loop, or (iii) B vanishes.

Now, let L be the length (part) of the loop for which B is tangential. Let Ie be the current enclosed by the loop. Then, Eq. ( .

) reduces to, BL =µ0Ie [ . (b)] When there is a system with a symmetry such as for a straight infinite current-carrying wire in Fig. . , the Ampere’s law enables an easy evaluation of the magnetic field, much the same way Gauss’ law helps in determination of the electric field.

This is exhibited in the Example . below. The boundary of the loop chosen is a circle and magnetic field is tangential to the circumference of the circle. The law gives, for the left hand side of Eq.

[ . (b)], B. 2πr. We find that the magnetic field at a distance r outside the wire is tangential and given by B × 2πr = µ0 I, B = µ0 I/ (2πr) ( .

) The above result for the infinite wire is interesting from several points of view. (i) It implies that the field at every point on a circle of radius r, (with the wire along the axis), is same in magnitude. In other words, the magnetic field possesses what is called a cylindrical symmetry. The field that normally can depend on three coordinates depends only on one: r.

Whenever there is symmetry, the solutions simplify. (ii) The field direction at any point on this circle is tangential to it. Thus, the lines of constant magnitude of magnetic field form concentric circles. Notice now, in Fig.

. (c), the iron filings form concentric circles. These lines called magnetic field lines form closed loops. This is unlike the electrostatic field lines which originate from positive charges and end at negative charges.

The expression for the magnetic field of a straight wire provides a theoretical justification to Oersted’s experiments. (iii) Another interesting point to note is that even though the wire is infinite, the field due to it at a nonzero distance is not infinite. It tends to blow up only when we come very close to the wire. The field is directly proportional to the current and inversely proportional to the distance from the (infinitely long) current source.

where I refers to the current passing through S. The sign of I is determined from the right-hand rule. We have discussed a simplified form of this law. If B is directed along the tangent to every point on the perimeter L of a closed curve and is constant in magnitude along perimeter then, BL = µ0 Ie where Ie is the net current enclosed by the closed circuit.

. The magnitude of the magnetic field at a distance R from a long, straight wire carrying a current I is given by: I B µ The field lines are circles concentric with the wire. . The magnitude of the field B inside a long solenoid carrying a current I is B = µ0nI where n is the number of turns per unit length.

For a toroid one obtains, NI B µ where N is the total number of turns and r is the average radius. . Parallel currents attract and anti-parallel currents repel. .

A planar loop carrying a current I, having N closely wound turns, and an area A possesses a magnetic moment m where, m = N I A and the direction of m is given by the right-hand thumb rule : curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of m (and A) When this loop is placed in a uniform magnetic field B, the force F on it is: F = And the torque on it is, ττ = m × B In a moving coil galvanometer, this torque is balanced by a counter- torque due to a spring, yielding kφ = NI AB where φ is the equilibrium deflection and k the torsion constant of the spring. . An electron moving around the central nucleus has a magnetic moment µl given by: l e l µ = where l is the magnitude of the angular momentum of the circulating electron about the central nucleus.

The smallest value of µl is called the Bohr magneton µB and it is µB = . × – J/T . A moving coil galvanometer can be converted into a ammeter by introducing a shunt resistance rs, of small value in parallel. It can be converted into a voltmeter by introducing a resistance of a large value in series.

Physical Quantity Symbol Nature Dimensions Units Remarks Permeability of free µ0 Scalar [MLT –2A– ] T m A– 4π × – T m A– space Magnetic Field B Vector [M T –2A– ] T (telsa) Magnetic Moment Vector [L2A] A m2 or J/T Torsion Constant k Scalar [M L2T – ] N m rad– Appears in MCG POINTS TO PONDER . Electrostatic field lines originate at a positive charge and terminate at a negative charge or fade at infinity. Magnetic field lines always form closed loops. .

The discussion in this Chapter holds only for steady currents which do not vary with time. When currents vary with time Newton’s third law is valid only if momentum carried by the electromagnetic field is taken into account. . Recall the expression for the Lorentz force, F = q (v × B + E) This velocity dependent force has occupied the attention of some of the greatest scientific thinkers.

If one switches to a frame with instantaneous velocity v, the magnetic part of the force vanishes. The motion of the charged particle is then explained by arguing that there exists an appropriate electric field in the new frame. We shall not discuss the details of this mechanism. However, we stress that the resolution of this paradox implies that electricity and magnetism are linked phenomena (electromagnetism) and that the Lorentz force expression does not imply a universal preferred frame of reference in nature.

. Ampere’s Circuital law is not independent of the Biot-Savart law. It can be derived from the Biot-Savart law. Its relationship to the Biot-Savart law is similar to the relationship between Gauss’s law and Coulomb’s law.