Question

$\huge{\frak{heya!\: Physics\:Newtons}}$
❥q+ are two charges and the distance between them is 2a.
qo is another charge between them. Now, if qo is slightly displace along line the two charges towards any of the charges. Show that qo executes SHM . Find time period of such SHM.
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Answer 1

Question:-

q+ are two charges and the distance between them is 2a.

qo is another charge between them. Now, if qo is slightly displace along line the two charges towards any of the charges. Show that qo executes SHM . Find time period of such SHM.

${\red{\underline{\underline{\bold{Answer:-}}}}}$

two charges each of magnitude +q are placed at a fixed distance L apart . another charge +Q and mass m is placed at midpoint of the line joining the charges . Let charge +Q displaces x slightly along A.

A(+q).....(L/2)........(+Q).........(L/2).......B(+q)

when charge +Q displaces x slightly along A.

force experienced by +Q due to charge A , F = KqQ/(L/2 - x)²

and force experienced by +Q due to charge B = KqQ/(L/2 + x)²

now, net force experienced by +Q = KqQ/(L/2 - x)² - KqQ/(L/2 + x)²

= KqQ[ {(L/2 + x)² - (L/2 - x)²}/(L²/4 - x²)² ]

= KqQ[L(2x)/(L²/4 - x²)]

here, L >> x

so, L²/4 >> x² and L²/4 - x² ≈ L²/4

Fnet = 2KqQLx/(L²/4)² = 32KqQx/L³

here it is clear that, force is directly proportional to x and direction of force is just opposite to direction of displacement. [ as net force acting along B direction but displacement is along A ]

so, we can use, Fnet = mw²x

hence, mw²x=(32kqQ/L³)x

so, w²= 32kqQ/mL³

and w=√32kqQ/mL³

so, time period , T =2π√mL³/32kqQ

hope it helps you..