Electricity and Magnetism A circular coil (radius R and mass M) has N turns of w

Question

Electricity and Magnetism

A circular coil (radius R and mass M) has N turns of wire carrying current I_o. The coil is free to rotate about the Z axis, which lies in the plane of the coil and passes through its center. There is a constant magnetic field B - B_0I, where I is a unit vector in the direction of x axis. Initially the coil is in stable equilibrium with its magnetic moment parallel to the field. (a) What is the frequency for small angle oscillations in phi about the equilibrium position? (b) If the coil is released from rest at the small angle phi_0 = 15 degrees, what is its angular velocity as it swings through the equilibrium position?

Explanation / Answer

a)   force on coil = n* Io *A *Bo *cos

                            =   n* Io *R2 *Bo *cos

       As, F *R = I * acc

=> angular acceleration = 2n* Io *R *Bo *cos/M

=>   angular velocity = sqrt(4n* Io **R *Bo *cos/M)

=> frequency = sqrt(n* Io **R *Bo *cos/M)/

b) angular velocity = sqrt(4n* Io *0.2617*R *Bo *0.9659/M)

                                   = 1.7818*sqrt(n* Io *R *Bo/M)

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