Since the set of differential length elements form a closed polygon and their

Since the set of differential length elements form a

This preview shows page 76 - 79 out of 110 pages.

Since the set of differential length elements ?? form a closed polygon, and their vector sum is zero, i.e., ∮ ?? = 0 . The net magnetic force on a closed loop is ? ? = 0 . 7.4.Torque on a Current Loop What happens when we place a rectangular loop carrying a current I in the xy plane and switch on a uniform magnetic field ? = ??̂ which runs parallel to the plane of the loop, as shown in Figure 7.4.1(a)? Figure 7.4.1 (a) A rectangular current loop placed in a uniform magnetic field. (b) The magnetic forces acting on sides 2 and 4 . From Eq. 8.4.1, we see the magnetic forces acting on sides 1 and 3 vanish because the length vectors ? 1 = ??̂ and ? 3 = ??̂ are parallel and anti-parallel to ? and their cross products vanish. On the other hand, the magnetic forces acting on segments 2 and 4 are non-vanishing: { ? 2 = ?(−??̂) × (B?̂) = IaB? ̂ ? 4 = ?(??̂) × (B?̂) = −IaB? ̂ (7.4.1) with ? 2 pointing out of the page and ? 4 into the page. Thus, the net force on the rectangular loop is ? ??? = ? 1 +? 2 + ? 3 + ? 4 = 0 (7.4.2)
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77 as expected. Even though the net force on the loop vanishes, the forces ? 2 and ? 4 will produce a torque which causes the loop to rotate about the y -axis (Figure 7.4.2). The torque with respect to the center of the loop is 𝜏 = (− ? 2 ?̂) × ? 2 + ( ? 2 ?̂) × ? 4 = (− ? 2 ?̂) × (???? ̂ ) + ( ? 2 ?̂) × (−???? ̂ ) (7.4.3) = ( 𝐼??? 2 + 𝐼??? 2 ) ?̂ = ?????̂ = ????̂ Where A ab represents the area of the loop and the positive sign indicates that the rotation is clockwise about the y -axis. It is convenient to introduce the area vector ? = ?? ̂ where ? ̂ is a unit vector in the direction normal to the plane of the loop. The direction of the positive sense of ? ̂ is set by the conventional right-hand rule. In our case, we have ? ̂ = +? ̂ . The above expression for torque can then be rewritten as 𝜏 = ?? × ? (7.4.4) Notice that the magnitude of the torque is at a maximum when ? is parallel to the plane of the loop (or perpendicular to ? ). Consider now the more general situation where the loop (or the area vector ? ) makes an angle with respect to the magnetic field. Figure 7.4.2 Rotation of a rectangular current loop From Figure 7.4.2, the lever arms and can be expressed as: ? 2 = ? 2 (−???𝜃 ? ̂ + ???𝜃 ? ̂ ) = − ? 4 (7.4.5) and the net torque becomes 𝜏 = ? 2 × ? 2 + ? 4 × ? 4 = 2 ∙ ? 2 (−???𝜃?̂ + ???𝜃? ̂ ) × (IaB? ̂ ) (7.4.6) = IabB???𝜃? ̂ = ?? × ?
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78 For a loop consisting of N turns, the magnitude of the toque is 𝜏 = ???? ???𝜃 (7.4.7) The quantity ??? is called the magnetic dipole moment µ : µ ⃗ = ??? (7.4.8) Figure 7.4.3 Right-hand rule for determining the direction of µ The direction of μ is the same as the area vector A (perpendicular to the plane of the loop) and is determined by the right-hand rule (Figure 8.4.3). The SI unit for the magnetic dipole moment is ampere- meter 2 (A m 2 ). Using the expression for µ , the torque exerted on a current-carrying loop can be rewritten as 𝜏 = µ ⃗ × ? (7.4.9) The above equation is analogous to 𝜏 = ? × ? in Eq. (1.8.3), the torque exerted on an electric dipole moment ? in the presence of an electric field ? . Recalling that the potential energy for an electric dipole is ? = ? ∙ ? [see Eq. (1.8.7)], a similar form is expected for the magnetic case. The work done by an external agent to rotate the magnetic dipole from an angle 0 to is given by ?
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