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A positively charged ring is placed in the y-z plane with its center at origin and a negatively charged particle is placed at some distance on x axis. Will it execute SHM?
A positively charged ring is placed in the y-z plane with its center at origin and a negatively charged particle is placed at some distance on x axis. Will it execute SHM?


2 years ago

Susmita
425 Points
							Let the particle of charge q be placed at a height x from the centre of the circular ring.Take any point on the circumference of the ring.Let R be the radius of the ring.So distance of charge q from this point will be$r=\sqrt{R^2 +x^2}$If Q is charge on the ring,then potential at r is$\phi=\frac{Q}{4\pi \epsilon r}$And electric field $E=-\frac{d}{dx}\phi =\frac{Qx}{4\pi \epsilon (R^2+x^2)^\frac{3}{2}}$If R>>x,ie,if the charge q is placed too close to the centre of the loop then E will approximate to$E=-\frac{d}{dx}\phi =\frac{Qx}{4\pi \epsilon R^3}$Force on any charged particle is qE.So the equation of force will become $Or,m\frac{d^2x}{dt^2}=-\frac{qQx}{4\pi \epsilon R^3}$We are putting the negative sign as q is negative.$Or,\frac{d^2x}{dt^2}+\frac{qQ}{4\pi \epsilon m R^3}x=0$$Or,\frac{d^2x}{dt^2}+(\omega)^2 x=0$Where$\omega = \frac{qQ}{4\pi \epsilon m R^3}$So it is equation of SHM.Yes the charge will execute SHM with angular frequency w.

2 years ago
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• 110 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions