A horizontal wire is hung from the ceiling of a room by two massless strings. Th

Question

A horizontal wire is hung from the ceiling of a room by two massless strings. The wire has a length of 0.18 m and a mass of 0.009 kg. A uniform magnetic field of magnitude 0.075 T is directed from the ceiling to the floor. When a current of I = 30 A exists in the wire, the wire swings upward and, at equilibrium, makes an angle ? with respect to the vertical, as the drawing shows. Find (a) the angle and (b) the tension in each of the two strings.

A horizontal wire is hung from the ceiling of a room by two massless strings. The wire has a length of 0.18 m and a mass of 0.009 kg. A uniform magnetic field of magnitude 0.075 T is directed from the ceiling to the floor. When a current of I = 30 A exists in the wire, the wire swings upward and, at equilibrium, makes an angle ? with respect to the vertical, as the drawing shows. Find (a) the angle and (b) the tension in each of the two strings.

Explanation / Answer

Draw a free body diagram about the wire. Looking down the wire we have weight (m*g) acting down, The FB (force from the magnetic field) to the rt = I*L*B and the tension T in the strings acting along the strings. (Due to symmetry the tension in each string is the same)

Now sum forces in the vertical we have 2*T*cos(?) = m*g
And in the horizontal I*L*B = 2*T*sin(?)

So dividing the 2nd eqn by the first to eliminate T

we get sin(?)/cos(?) = tan(?) = I*L*B/m*g = 30*0.18*0.075/(0.009*9.8) = 4.59183

so ? = arctan(4.59183) = 77.71o

b) therefore the tension in each string is T = m*g/(2*cos(?)) = 0.009*9.8/(2*cos(77.71)) = 0.20717N

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