Simple Nature - Light and Matter

6 Two parallel wires of length L carry currents I 1 and I 2 . Theyare separated by a distance R, and we assume R is much less thanL, so that our results for long, straight wires are accurate. The goalof this problem is to compute the magnetic forces acting betweenthe wires.(a) Neither wire can make a force on itself. Therefore, our first stepin computing wire 1’s force on wire 2 is to find the magnetic fieldmade only by wire 1, in the space occupied by wire 2. Express this√field in terms of the given quantities.(b) Let’s model the current in wire 2 by pretending that there isa line charge inside it, possessing density per unit length λ 2 andmoving at velocity v 2 . Relate λ 2 and v 2 to the current I 2 , using theresult of problem 5a. Now find the magnetic force wire 1 makes onwire 2, in terms of I 1 , I 2 , L, and R. ⊲ Answer, p. 930(c) Show that the units of the answer to part b work out to benewtons.7 Suppose a charged particle is moving through a region ofspace in which there is an electric field perpendicular to its velocityvector, and also a magnetic field perpendicular to both the particle’svelocity vector and the electric field. Show that there will be oneparticular velocity at which the particle can be moving that resultsin a total force of zero on it. Relate this velocity to the magnitudesof the electric and magnetic fields. (Such an arrangement, called avelocity filter, is one way of determining the speed of an unknownparticle.)8 The following data give the results of two experiments in whichcharged particles were released from the same point in space, andthe forces on them were measured:q 1 = 1 µC , v 1 = (1 m/s)ˆx , F 1 = (−1 mN)ŷq 2 = −2 µC , v 2 = (−1 m/s)ˆx , F 2 = (−2 mN)ŷThe data are insufficient to determine the magnetic field vector;demonstrate this by giving two different magnetic field vectors, bothof which are consistent with the data.9 The following data give the results of two experiments in whichcharged particles were released from the same point in space, andthe forces on them were measured:q 1 = 1 nC , v 1 = (1 m/s)ẑ , F 1 = (5 pN)ˆx + (2 pN)ŷq 2 = 1 nC , v 2 = (3 m/s)ẑ , F 2 = (10 pN)ˆx + (4 pN)ŷIs there a nonzero electric field at this point? A nonzero magneticfield?10 This problem is a continuation of problem 6. Note that theanswer to problem 6b is given on page 930.(a) Interchanging the 1’s and 2’s in the answer to problem 6b, whatis the magnitude of the magnetic force from wire 2 acting on wire1? Is this consistent with Newton’s third law?718 Chapter 11 Electromagnetism