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A block is attached to the ceiling by a spring that has a force constant $k = 200 N / m$ . A conducting rod is rigidly attached to the block. The combined mass of the block and the rod is $m = 0.3 k g$ . The rod can slide without friction along two vertical parallel rails, which are a distance $L = 1 m$ apart. A capacitor of known capacitance $C = 500 μ F$ is attached to the rails by the wires. The entire system is placed in a uniform magnetic field $B = 20 T$ directed as shown. Find the angular frequency (in rad/sec) of the vertical oscillations of the block. Neglect the self -inductance and electrical resistance of the rod and all wires

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Detailed Solution

Total energy, E = $\frac{1}{2} m v^{2} + \frac{1}{2} k x^{2} + \frac{1}{2} \frac{q^{2}}{C}, e = B v l = \frac{q}{C}$

$E = \frac{1}{2} (m + B^{2} I^{2} C) v^{2} + \frac{1}{2} k x^{2}$ as E=constant $\frac{d E}{d t} = 0, \frac{d v}{d t} = - (\frac{k}{m + B^{2} I^{2} C}) x \propto x \Rightarrow S H M$

$ω = \sqrt{\frac{k}{m + B^{2} I^{2} C}} = 20 r a d / \sec$

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