In the given figure ABC is a $$non-conducting$$ $$semi-circular$$ wire of radius ‘a’ carrying a total charge Q uniformly distributed on it and a point charge q is at its centre. Ends of the wire are attached to two separate springs each having spring constant k as shown in the figure. In the given position system is in equilibrium. Now the point charge is suddenly removed. The amplitude of the resulting oscillation is $$xqQ8$$$$π$$$$2$$ε$$0a2k$$. Ignoring any force other than spring force and electrostatic force find the value of ‘x’?

Question

In the given figure ABC is a  $$non-conducting$$ $$semi-circular$$ wire of radius ‘a’ carrying a total charge Q uniformly distributed on it and a point charge q is at its centre. Ends of  the wire are attached to two separate springs each having spring constant k as shown in the figure. In the given position system is in equilibrium. Now the point charge is suddenly removed. The amplitude of the resulting oscillation is  $$xqQ8$$$$π$$$$2$$ε$$0a2k$$. Ignoring any force other than spring force and electrostatic force find the value of ‘x’?

Infinity Learn · Accepted Answer

$$F=$$∫dFcos⁡θ$$=$$∫$$14$$πε$$0q$$λ$$R(d$$θ$$)R2cos$$⁡θ$$=14$$πε$$0q$$λ$$R($$$$sin$$⁡θ$$)$$−π$$/2$$π$$/2=2q$$λ$$4$$πε$$0R$$⇒$$2kA=2q$$λ$$4$$πε$$0R$$⇒$$kA=q4$$πε$$0RQ$$$$π$$R⇒$$A=Qq4$$$$π$$$$2$$ε$$0a2k$$∵$$R=a$$⇒$$A=2Qq8$$$$π$$$$2$$ε$$0a2k$$⇒$$x=2$$

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