MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2007 Problem Set 8 Solutions Problem 1: Charges in a Uniform Field An electron with velocity ˆˆxyvv+ijmoves through a uniform magnetic field ˆˆxyBB+ij(a) Find the magnitude of the force on the electron. (b) Repeat your calculation for a proton having the same velocity. This is just a question of calculating a cross-product. Although it wasn’t introduced in class, the way to do this is with a determinant: ()ˆˆˆˆ00xyxyyxxyqq vvq v Bv BBB=×==−ijkFvBGGGkSo the magnitude of the force is ()xyyxFe v Bv B=−for both an electron and proton, the only difference is the direction is reversed (q=-efor an electron). Problem 2: Power Line A horizontal power line carries a current of Ifrom south to north. Earth's magnetic field (BEarth= 62.1 μT) is directed toward the north and is inclined downward at an angle θto the horizontal. Find the magnitude of the magnetic force on a length Lof the line due to Earth's field. Please also calculate a value numerically for I= 3000 A, θ= 78.0º and L= 100 m, since it’s useful to think about real world numbers. Since the line is running south to north, it has an angle θrelative to the Earth’s field. Thus: ()sinIFILBθ=×⇒=FLBGGGFor the numbers given, ()()()()()sin3000 A100 m62.1 sin78º18 NFILBTθμ===, which is a small force. For comparison, if the cable has a mass of 300 kg (about what a one cm radius copper wire of that length will weigh) it will feel a force of about 3000 N due to gravity. Of course most power lines are AC so this will also be an oscillating, not unidirectional force. Problem 3: Triangular Loop A current loop, carrying a current I, is in the shape of a right triangle with sides 30, 40, and 50 cm. The loop is in a uniform magnetic field Bwhose direction is parallel to the current in the 50 cm side of the loop. Find the magnitude of (a)the magnetic dipole moment of the loop in amperes-square meters and (b)the torque on the loop. The loop has an area of 212baseheight600 cmA=⋅=so its dipole moment is (a) 21200cmIμ=⋅(b) The torque is 2600cm(in the plane of the loop, perpendicular to the hypotenuse)IB=×=τμBGGGPS08 Solutions-1
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