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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? 

Grade:12th pass

1 Answers

Susmita
425 Points
3 years ago
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.
 

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