Total charge -Q is uniformly spread along the length of a ring of radius R. A small test charge +q of mass m is kept at the centre of the ring and is given a gentle push along the axis of the ring.
(a) Show that the particle executes a simple harmonic oscillation.
(b) Obtain its time period.

This is a Long Answer Type Questions as classified in NCERT Exemplar

Explanation- electric field at the axis of the ring is E=KQy/ (R²+z²)^3/2 where z is distance .

F=qE=KQqy/ (R²+z²)^3/2

When z<

F= $\frac{2\pi }{w}$ = -Kz

So force is directly proportional to – distance that is completely defined that is follows S.H.M

(b)w= $\pi \sqrt[]{\frac{m}{k}}$

T=2 $\pi \sqrt[]{\frac{m4\pi {\epsilon }_o{d}^3}{2q^2}}$ w=2 $\frac{kqQy}{R3}$

T=2 $\surd \frac{k}{m}$

<p>This is a Long Answer Type Questions as classified in NCERT Exemplar</p><p><strong>Explanation-</strong> electric field at the axis of the ring is E=KQy/ (R²+z²)^3/2where z is distance .</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1741774919phpLB38Yd_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1741774919phpLB38Yd.jpeg" alt="" width="155" height="108"></picture></div><div><p>F=qE=KQqy/ (R²+z²)^3/2</p><p>When z&lt;&lt;r&lt; p=""&gt; &lt;/r&lt;&gt;</p><p>F=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </mfrac> </math> </span>= -Kz</p><p>So force is directly proportional to – distance that is completely defined that is follows S.H.M</p><p> (b)w=<span contenteditable="false"> <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span></p><p>T=2<span contenteditable="false"> <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>m</mi> <mn>4</mn> <mi>π</mi> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>d</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <msup> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span> w=2<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>k</mi> <mi>q</mi> <mi>Q</mi> <mi>y</mi> </mrow> </mrow> <mrow> <mrow> <mi>R</mi> <mn>3</mn> </mrow> </mrow> </mfrac> </math> </span></p><p>T=2<span contenteditable="false"> <math> <mo>√</mo> <mfrac> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </mfrac> </math> </span></p></div></div>

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