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Figure below shows a cantilever type scanner that is actuated by using electromagnetic force/torque. There is a single-loop coil with negligible pitch to the edge of the mirror (i.e. rectangular coil width and length can be assumed as W and D). The spring can produce out-of-plane (call it mode-1) and torsional (call it mode-2) displacements around the suspension. Use Silicon as the structural material which has a uniform thickness of t, and a density of p,ignore the mass of the spring. Spring width is a and spring length is L. Mechanical quality factors of mode-1 andmode-2 are Q1 and Q1, respectively. You may denote the necessary material constants with subscript Si, in case you need to use them (e.g. Young's modulus of Silicon Esi).

(a) Find expressions for resonance frequencies of out-of-plane motion and torsion about spring length direction. (b) When an external, in-plane B-field of 1 Tesla is imposed with a 45 degree inclination to the mirror plate, what would be the "resonant" rotation angles for mode-1 and mode2 if we pass an ac current with a magnitude ofI? (b) When an external, in-plane B-field of 1 Tesla is imposed with a 45 degree inclination to the mirror plate, what would be the "resonant" rotation angles for mode-1 and mode2 if we pass an ac current with a magnitude ofI?

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