2 821 86 Clearly 12 points toward wire 2 The conclusion we can draw from this

2 821 86 clearly 12 points toward wire 2 the

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2?? ? ̂ (8.2.1)
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86 Clearly ? 12 points toward wire 2. The conclusion we can draw from this simple calculation is that two parallel wires carrying currents in the same direction will attract each other. On the other hand, if the currents flow in opposite directions, the resultant force will be repulsive. 8.3. Ampere’s Law We have seen that moving charges or currents are the source of magnetism. This can be readily demonstrated by placing compass needles near a wire. As shown in Figure 8.3.1a, all compass needles point in the same direction in the absence of current. However, when I 0, the needles will be deflected along the tangential direction of the circular path (Figure 8.3.1b). Figure 8.3.1 Deflection of compass needles near a current-carrying wire Let us now divide a circular path of radius r into a large number of small length vectors ∆? = ∆?? ̂ , that point along the tangential direction with magnitude ∆? (Figure 8.3.2). Figure 8.3.2 Amperian loop In the limit ∆? → 0 we obtain ∮ ? ∙ ?? = ? ∮ ?? = ? 0 𝐼 2?? (8.3.1)
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87 The result above is obtained by choosing a closed path, or an “Amperian loop” that follows one particular magnetic field line. Let’s consider a slightly more complicated Am perian loop, as that shown in Figure 8.3.3 Figure 8.3.3 An Amperian loop involving two field lines The line integral of the magnetic field around the contour abcda is ? ∙ ?? ? ????? = ∮ ? ∙ ?? ? ?? + ∮ ? ∙ ?? ? ?? + ∮ ? ∙ ?? ? ?? + ∮ ? ∙ ?? ? ?? = 0 + ? 2 (? 2 𝜃) + ? 1 [? 1 (2? − 𝜃)] (8.3.2) where the length of arc bc is ? 2 𝜃 , and ? 1 (2? − 𝜃) for arc da . The first and the third integrals vanish since the magnetic field is perpendicular to the paths of integration. With B 1 0 I / 2 r 1 and B 2 0 I / 2 r 2 , the above expression becomes ? ∙ ?? ? ????? = ? 0 𝐼 2?? 2 (? 2 𝜃) + ? 0 𝐼 2?? 1 [? 1 (2? − 𝜃)] = ? 0 𝐼 2? 𝜃 + ? 0 𝐼 2? (2? − 𝜃) (8.3.3) We see that the same result is obtained whether the closed path involves one or two magnetic field lines. As shown in Example 9.1, in cylindrical coordinates ( r , , z ) with current flowing in the + z -axis, the magnetic field is given by ? ( 0 I / 2 r ) φ ˆ . An arbitrary length element in the cylindrical coordinates can be written as ?? = ???̂ + ???? ̂ + ???̂ (8.3.4) which implies ? ∙ ?? ? ?????? ???ℎ = ∮ ( ? 0 𝐼 2? )??? 0 ?????? ???ℎ = ( ? 0 𝐼 2? )∮ ?? 0 ?????? ???ℎ = ? 0 𝐼 2? (2?) = ? 0 ? (8.3.5)
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88 In other words, the line integral of ∮ ? ∙ ?? around any closed Amperian loop is proportional to ? ??? , the current encircled by the loop. Figure 8.3.4 An Amperian loop of arbitrary shape. The generalization to any closed loop of arbitrary shape (see for example, Figure 9.3.4) that involves many magnetic field lines is known as Ampere’s law: ∮ ? ∙ ?? = ? 0 ? ??? (8.3.6) Ampere’s law in magnetism is analogous to Gauss’s law in electrostatics. In order to apply them, the system must possess certain symmetry. In the case of an infinite wire, the system possesses cylindrical symmetry and Ampere’s law can be readily applied. However, when the length of the wire is finite, Biot-Savart law must be used instead. Biot-Savart Law ? = ? 0 ? 4? ?? × ?̂ ? 2 general current source ex: finite wire Ampere’s law ∮ ? ∙ ?? = ? 0 ?
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