I\'ve gotten this answer wrong twice (once using a posted solution), so a clear

Question

I've gotten this answer wrong twice (once using a posted solution), so a clear solution would be helpful. Thanks

Two long, parallel wires, each with a mass per unit length of 91 g/m, are supported in a horizontal plane by 6.0-cm-long strings, as shown in the figure below. Each wire carries the same current I causing the wires to repel each other so that the angle 0 between the supporting strings is 16 6.0 cm (a) Are the currents in the same or opposite directions? same direction opposite direction O not enough information to decide (b) Determine the magnitude of each current. A

Explanation / Answer

Magnetic Force per unit length between parallel wires is given by

F/l = u0 I^2/2 pi d

Let, T be the tension in the strings supporting the wires. Force for the given wires in X and Y directions will be:

Fx = T sin(theta) - F = 0

Fy = T cos(thets) - mg = 0

F = T sin(theta) (1)

T = mg/cos(theta) (2)

putting the value of T from (2) into (1)

F = mg sin(theta)/cos(theta) = mg tan(theta)

dividing by l

F/l = (m/l)g tan(theta) = g lambda tan(theta) [where, lambda = m/l]

u0 I^2/ 2 pi d = g lambda tan(theta)

the distance between the wires using trigonometry is:

d = 2 h sin(theta)

u0 I^2/ 2 pi ( 2 h sin(thet)) = g lambda tan(theta)

Solving for I we get

I = sqrt [ (4 pi/ u0) g lambda h sin(theta) tan(theta)]

I = sqrt [ (4pi/ 4pi x 10^-7) 9.8 x 0.06 x 91 x 10^-3 sin16 x tan16]

I = sqrt [ (9.8 x 0.06 x 91 x 10^-3 x 0.276 x 0.288/10^-7)] = 206.23 A

Hence, I = 206.23 A

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