2011 Semifinal Exam Part B Part B Question B1 An AC power line cable transmits electrical power using a sinusoidal waveform with frequency 60 Hz.The load receives an RMS voltage of 500 kV and requires 1000 MW of average power. For this problem,consider only the cable carrying current in one of the two directions,and ignore effects due to capacitance or inductance between the cable and with the ground. a.Suppose that the load on the power line cable is a residential area that behaves like a pure resistor. i.What is the RMS current carried in the cable? ii.The cable has diameter 3 cm,is 500 km long,and is made of aluminum with resistivity 2.8 x 10-8 n.m.How much power is lost in the wire? b.A local rancher thinks he might be able to extract electrical power from the cable using electromagnetic induction.The rancher constructs a rectangular loop of length a and width b<a,consisting of N turns of wire.One edge of the loop is to be placed on the ground;the wire is straight and runs parallel to the ground at a height h much less than the length of the wire.Write the current in the wire as I =Io sinwt,and assume the return wire is far away. i.Determine an expression for the magnitude of the magnetic field at a distance r from the power line cable in terms of I,r,and fundamental constants. ii.Where should the loop be placed,and how should it be oriented,to maximize the induced emf in the loop? iii.Assuming the loop is placed in this way,determine an expression for the emf induced in the loop (as a function of time)in terms of any or all of lo,h,a,b,N,w,t,and fundamental constants. iv.Suppose that a =5 m,b=2 m,and h=100 m.How many turns of wire N does the rancher need to generate an RMS emf of 120 V? c.The load at the end of the power line cable changes to include a manufacturing plant with a large number of electric motors.While the average power consumed remains the same,it now behaves like a resistor in parallel with a 0.25 H inductor. i.Does the power lost in the power line cable increase,decrease,or stay the same?(You need not calculate the new value explicitly,but you should show some work to defend your answer.) ii.The power company wishes to make the load behave as it originally did by installing a capacitor in parallel with the load.What should be its capacitance? Copyright C2011 American Association of Physics Teachers