The unit of magnetic flux is named for Wilhelm Weber. The practical-size unit of

Question

The unit of magnetic flux is named for Wilhelm Weber. The practical-size unit of magnetic field is named for Johann Karl Friedrich Gauss. Both were scientists at Goltingern, Germany, in addition to their individual accomplishments, they built a telegraph together in 1833. It consisted of a battery and switch that were positioned at one end of a transmission line 3 km long operating an electromagnetic at the other end. (Andre Ampere suggested electrical signal in 1821. Samuel Morse built a telegraph line between Baltimore and Washington in 1844.) Suppose that Weber and Gauss's transmission line was as diagrammed in the figure below. Two long, parallel wires, each having a mass per unit length of 34.0 g/m, are supported in a horizontal plane by strings 5.50 cm long. Each wire carries the same current i, causing the wires to repel each other so that the angle theta between the supporting strings is 12.0 degree. (a) Are the currents in the same or in opposite directions? Answer same or opposite _________ (b) Find the magnitude of the current. Round your answer to three significant figures. Take the permeability of free space to be 4 pi times 10^-7 T middot m/A, the free fall acceleration to be 9.8 m/s^2 __________ A

Explanation / Answer

mass per unit length of the wires, lambda = 34g/m
length of strings, l = 5.5 cm
current in wire = i ( each wire)
theta = 12 degree, so angle with the vertical of any string = theta/2 = 6 degree

a. as the wires are being repelled and we know that like currents attract, hence the currents are in opposite directions
b. now, from force balance, assuming tension in the string be T, and magnetic force on wire be F
   Tcos(theta/2) = mg ( m is mass of wire)
   Tsin(theta/2) = F
   tan(theta/2) = F/mg
   but m = lambda*L [ where lambdais mass per unit length of the wire asnd l is te length of the wire]
   also, F/L = 2ki^2/x [ x is seperation between wires]
   from trigonometry
   sin(theta/2) = x/2l, x = 2l*sin(theta/2)
   so, tan(theta/2) = 2ki^2*L/2lsin(theta/2)*lambda*L = ki^2/lsin(theta/2)*lambda = 10^-7*i^2/0.055*sin(6)*0.034
   i = 14.33 A

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