# Three point charges of +2 uC , -3uC , and -3uC are kept at the vertices A,B and C respectively of an equilateral triangle of side 20 cm as shown in the figure(attached with question).What should be the sign and magnitude of the charge to be placed at the midpoint M of the side BC so that the charge at A remain in equilibruium. (Delhi -2005)

1
by arnavsahay803

2015-04-27T00:14:31+05:30

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See diagram.

Let K = 1/(4πε) = 9 * 10⁹ N-m²/C²

The forces F1 and F2 on the charge at Q are along the sides AB and AC.
magnitude of F1 = F2 = K * 2 * 10⁻⁶ * 3 * 10⁻⁶ / 0.20²
=  1.5 * 10⁻⁸ K Newtons

Their components along the horizontal directions are  equal to F1 Cos 60° and F2 Cos 60° which are equal in magnitude and opposite in direction. They cancel.

Their components along AM will add to :
2 F1 Cos 30° = 2 * 1.5 * 10⁻⁸ * √3/2 * K  newtons  = 1.5√3 *10⁻⁸ K  Newtons

Let the charge placed at M be q Coulombs.  AM² = 0.20² - 0.10² = 0.03 m²

Force due to q on charge at A will be along MA:  F3
F3 = K q * 2 * 10⁻⁶ / 0.03 = 6.667 q K * 10⁻⁹  Newtons

For the charge at A to be stationary,

6.667 q K * 10⁻⁹ = 1.5 √3 * 10⁻⁸ K

q = 3.897  Coulombs  positive charge.