A rectangular loop rotates with a constant angular velocity about an axis as sho

Question

A rectangular loop rotates with a constant angular velocity about an axis as shown. It makes one complete rotation in 0.05 s. The magnetic field lines are straight and parallel as shown. They are perpendicular to the axis of rotation. The lengths are c= 0.6 m and d= 3.9 m, the strength of the magnetic field is 0.1 T. The total resistance of the loop is 170 W.

(a) What is the magnetic flux through the loop at the instant the normal to the loop makes an angle of 0° with the magnetic field?
(b) What is the magnetic flux through the loop at the instant the normal to the loop makes an angle of 30° with the magnetic field?
(c) What is the magnetic flux through the loop at the instant the normal to the loop makes an angle of 90° with the magnetic field?
(d) What is the angle between the normal to the loop and the B-field when the emf in the loop is at its maximum value?
(e) What is the maximum current I through the loop?
(f) At the instant when the current is a maximum, what torque (magnitude) must be applied by outside forces to keep the loop rotating at a constant angular velocity?

Explanation / Answer

Given magnitude of magnetic field   B = 0.1 T Area of rectangular loop A = 0.6 * 3.9 m2 = 2.34 m 2   resistance of loop R = 170 time taken to complet one rotation   t = 0.05 s       a) magnetic flux through the loop makes an angle 0 degrees                   B   = BA cos                          = (0.1)(2.34) cos 0                           = 0.234 Wb     b) magnetic flux through the loop makes an angle 30 degrees with the magnetic field                     B   = BA cos                          = (0.1)(2.34) cos 30                           = 0.2 Wb      c) magnetic flux through the loop makes an angle 90 degrees with the magnetic field                     B   = BA cos                          = (0.1)(2.34) cos 90                           = 0   Wb     d)   induced emf in the loop is                     = d / dt                        = d/ dt (BA cos)          emf in the loop has maximum value at 0 degrees     e) induced emf in the loop is                         = 0.234    / 0.05                          = 4.68 V           maximum current through the loop is                    I = / R                        = 4.68 / 170                        = 0.0275 A    f)   torque on the loop                = N iAB                    = 0.0275 * 2.34 * 0.1                    = 0.00644 N m                          = (0.1)(2.34) cos 30                           = 0.2 Wb      c) magnetic flux through the loop makes an angle 90 degrees with the magnetic field                     B   = BA cos                          = (0.1)(2.34) cos 90                           = 0   Wb     d)   induced emf in the loop is                     = d / dt                        = d/ dt (BA cos)          emf in the loop has maximum value at 0 degrees     e) induced emf in the loop is                         = 0.234    / 0.05                          = 4.68 V           maximum current through the loop is                    I = / R                        = 4.68 / 170                        = 0.0275 A    f)   torque on the loop                = N iAB                    = 0.0275 * 2.34 * 0.1                    = 0.00644 N m                     B   = BA cos                          = (0.1)(2.34) cos 90                           = 0   Wb     d)   induced emf in the loop is                     = d / dt                        = d/ dt (BA cos)          emf in the loop has maximum value at 0 degrees     e) induced emf in the loop is                         = 0.234    / 0.05                          = 4.68 V           maximum current through the loop is                    I = / R                        = 4.68 / 170                        = 0.0275 A    f)   torque on the loop                = N iAB                    = 0.0275 * 2.34 * 0.1                    = 0.00644 N m                          = (0.1)(2.34) cos 90                           = 0   Wb     d)   induced emf in the loop is                     = d / dt                        = d/ dt (BA cos)          emf in the loop has maximum value at 0 degrees     e) induced emf in the loop is                         = 0.234    / 0.05                          = 4.68 V           maximum current through the loop is                    I = / R                        = 4.68 / 170                        = 0.0275 A    f)   torque on the loop                = N iAB                    = 0.0275 * 2.34 * 0.1                    = 0.00644 N m

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