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# Rotating charge

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A charge of 4.0 x 10^-6 C is placed on a small conducting sphere that is located at the end of a thin insulating rod whose length is 0.20 m. The rod rotates with an angular speed of 150 radians/second about an axis that passes perpendicularly through its other end. What is the magnetic moment of the rotating charge?

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A charge of 4.0 x 10-6 C is placed on a small conducting sphere that is located at the end of a thin insulating rod whose length is 0.20 m. The rod rotates with an angular speed of 150 radians/second about an axis that passes perpendicularly through its other end. What is the magnetic moment of the rotating charge?

Solution :

Q = 4x10-6 C
0.2m

The charge is moving with a constant speed on a circular path. The frequency of the circular motion (i.e. number of times the charge passes by any given point per second) f is related to the angular speed ω as : ω = 2Πf. Substituting value of ω we get :

150 = 2Πf or f = 150/2Π = 23.88/sec

As moving charge constitutes electric current, the current I in this case is defined as :

Current I = dQ/dt = Rate of flow of charge = Total charge passing through a given point per sec = Charge of the sphere x frequency of rotation = 4x10-6x23.88 = 95.52 x 10-6 Ampere

The above situation is equivalent to a current carrying circular loop carrying a current of 95.52 x 10-6 A. Magnetic moment of a current carrying circular loop is defined as :
M = AI where A = Area of the loop and I = Current

Hence, M = AI = (ΠR2)I = Π x 0.22 x 95.52 x 10-6 = 12 x 10-6 Am2

Magnetic moment is a vector quantity. In the present case its magnitude is as calculated above and its direction is perpendicular to the plane of rotation of the rod. The sense (i.e. out of the plane or into the plane) is given by the right hand rule which states : curve the fingers of the right hand such that they point in the direction of the current flow. Then, the stretched out thumb gives the sense of the magnetic moment vector. If the direction of rotation of the rod is clockwise, the magnetic moment vector arrow points into the plane as seen from top and for counterclockwise in points out of the plane.

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