4.4 A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line?

4.6 AMPERE’S CIRCUITAL LAW There is an alternative and appealing way in which the Biot-Savart law may be expressed. Ampere’s circuital law considers an open surface with a boundary (Fig. 4.12). The surface has current passing through it. We consider the boundary to be made up of a number of small line elements. Consider one such element of length dl. We take the value of the tangential component of the FIGURE 4.12 117magnetic field, Bt, at this element and multiply it by the Reprint 2025-26 Physics length of that element dl [Note: Btdl=B.dl]. All such products are added together. We consider the limit as the lengths of elements get smaller and their number gets larger. The sum then tends to an integral. Ampere’s law states that this integral is equal to µ0 times the total current passing through the surface, i.e., “B.dl = µ0I [4.13(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 Andre Ampere (1775 – current I is regarded as positive. 1836) Andre Marie Ampere For several applications, a much simplified version of was a French physicist, mathematician and chemist Eq. [4.13(a)] proves sufficient. We shall assume that, in who founded the science of such cases, it is possible to choose the loop (called electrodynamics. Ampere an amperian loop) such that at each point of the was a child prodigy loop, either who mastered advanced (i) B is tangential to the loop and is a non-zero constant mathematics by the age of B, or 12. Ampere grasped the significance of Oersted’s (ii) B is normal to the loop, or discovery. He carried out a (iii) B vanishes. large series of experiments Now, let L be the length (part) of the loop for which B to explore the relationship is tangential. Let Ie be the current enclosed by the loop. between current electricity Then, Eq. (4.13) reduces to, and magnetism. These investigations culminated BL =µ0Ie [4.13(b)] in 1827 with the When there is a system with a symmetry such as for publication of the a straight infinite current-carrying wire in Fig. 4.13, the ‘Mathematical Theory of Ampere’s law enables an easy evaluation of the magnetic–1836) Electrodynamic Pheno- mena Deduced Solely from field, much the same way Gauss’ law helps in Experiments’. He hypo- determination of the electric field. This is exhibited in the thesised that all magnetic Example 4.8 below. The boundary of the loop chosen is(1775 phenomena are due to a circle and magnetic field is tangential to the circulating electric circumference of the circle. The law gives, for the left hand currents. Ampere was side of Eq. [4.13 (b)], B. 2πr. We find that the magnetic humble and absent- field at a distance r outside the wire is tangential and minded. He once forgot anAMPERE given by invitation to dine with the Emperor Napoleon. He died B × 2πr = µ0 I, of pneumonia at the age of 61. His gravestone bears B = µ0 I/ (2πr) (4.14)ANDRE the epitaph: Tandem Felix The above result for the infinite wire is interesting (Happy at last). 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 118 magnitude. In other words, the magnetic field Reprint 2025-26 Moving Charges and Magnetism 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. 4.1(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 non-zero 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. (iv) There exists a simple rule to determine the direction of the magnetic field due to a long wire. This rule, called the right-hand rule*, is: Grasp the wire in your right hand with your extended thumb pointing in the direction of the current. Your fingers will curl around in the direction of the magnetic field. Ampere’s circuital law is not new in content from Biot-Savart law. Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current. Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law. Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge). We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time. The following example will help us understand what is meant by the term enclosed current. Example 4.7 Figure 4.13 shows a long straight wire of a circular cross-section (radius a) carrying steady current I. The current I is uniformly distributed across this cross-section. Calculate the magnetic field in the region r < a and r > a. EXAMPLE 4.7 FIGURE 4.13 * Note that there are two distinct right-hand rules: One which gives the direction of B on the axis of current-loop and the other which gives direction of B for a straight conducting wire. Fingers and thumb play different roles in 119 the two. Reprint 2025-26 Physics Solution (a) Consider the case r > a. The Amperian loop, labelled 2, is a circle concentric with the cross-section. For this loop, L = 2 π r Ie = Current enclosed by the loop = I The result is the familiar expression for a long straight wire B (2π r) = µ0I µ0 I B = 2 π r [4.15(a)] 1 B ∝ (r > a) r Now the current enclosed Ie is not I, but is less than this value. Since the current distribution is uniform, the current enclosed is,  π r 2  Ir 2 I e = I 2 a 2  π  = a I r 2 B (2 π r ) = µ0 2 Using Ampere’s law, a  µ0 I  B = 2 r [4.15(b)]  2πa  B ∝ r (r < a) FIGURE 4.14 Figure (4.14) shows a plot of the magnitude of B with distance r 4.7 from the centre of the wire. The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section. This example possesses the required symmetry so that Ampere’s EXAMPLE law can be applied readily. It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case. For example, for the case of the circular loop discussed in Section 4.5, it cannot be applied to extract the simple expression B = µ0I/2R [Eq. (4.12)] for the field at the centre of the loop. However, there exists a large number of situations of high symmetry where the law 120 can be conveniently applied. We shall use it in the next section to calculate Reprint 2025-26 Moving Charges and Magnetism the magnetic field produced by a commonly used and very useful magnetic system: the solenoid.

1.14 Consider a uniform electric field E = 3 × 103 î N/C. (a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?