Consider an equilateral triangle ABC of side 2a in the plane of the paper as shown. The centroid of the triangle is O. Equal charges (Q) are fixed at the vertices A, B and C. In what follows consider all motion and situations to be confined the plane of the paper.
) felt by the test charge.
) on this test charge if it is placed at the point D as shown in the figure.
Text Solution
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Here K = 1/4πε 0 and direction is upward (towards A)
Using binomial approximation, 
(upward) which is linear in δ . Hence charge will oscillate simple harmonically about O when released.
(downward)
For small δ force on the test charge is upwards while for large δ (eg. at D) force is downwards. So there is a neutral point between O and D. By symmetry there will be neutral points on other medians also. In figure x. Below all possible (4) neutral points are shown by ∙ .
(e) Let the distance along P be x and O to be at (0, 0). Electric potential of a test charge along OP can be written as
V(x) = 
≈ KQ 
We can see that V(x) ∝ x 2 , hence it is a stable equilibrium.
(f) Equilibrium points are indicated by ∙ .
(g) N + 1
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