A particle of mass M and charge q is at rest at the mid point between two other fixed similar charges each of magnitude Q placed a distance 2 d apart. The system is collinear as shown in the figure. T

Question

A particle of mass M and charge q is at rest at the mid point between two other fixed similar charges each of magnitude Q placed a distance 2 d apart. The system is collinear as shown in the figure. The particle is now displaced by a small amount x ( x << d ) along the joining the two charges and is left to itself. It will now oscillate about the mean position with a time period ( ε 0 ​ = permittivity of free space)

TestHub · Accepted Answer

Restoring force on displacement of x F F ​ = K [ ( d − x ) 2 q ​ − ( d + x ) 2 Qq ​ ] = K Qq [ ( d − x ) 2 1 ​ − ( d + x ) 2 1 ​ ] = K Qq [ ( d 2 − x 2 ) 2 4 d x ​ ] = K Qq [ d 4 4 d x ​ ] If ( d >> x ) = K Qq [ d 3 4 x ​ ] ​ Acceleration, a = m F ​ = M d 3 4 K Qq x ​ or ω 2 = M d 3 4 K Qq ​ ∴ T = ω 2 π ​ = 2 π 4 K Oq M d 3 ​ ​ = 2 Qq π 3 M d 3 ε 0 ​ ​ ​

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