Q.2. Total electrostatic potential energy of a system of charges

Question 2:

Six charges of magnitude +q and -q are fixed at the corners of a regular hexagon of edge length a as shown in the figure.The electrostatic interaction energy of the charged particles is:

Answer 2:

Let the vertices represented by 1, 2, 3, 4, 5 and 6 have charges

+q, - q, +q, - q, + q and - q.

2 out of 6 vertices can be selected in 6C2 ways = 15 ways

=> we have to find PE due to these 15 combination of charges

PE due to charge-combinations, 1-2, 2-3, 3-4, 4-5, 5-6 and 6-1

= - 6q^2 / 4pi*a*εo

PE due to charge combinations, 1-3, 1-5, 2-4, 2-6, 3-5 and 4-6

= + 6q^2 / 4pi*(a√3)*εo

[Note: distance between these charges is a√3]

PE due to charge combinations, 1-4, 2-5 and 3-6

= - 3q^2 / 4pi*(2a)*εo

[Note: distance between these charges is 2a]

Total

= (q^2)/(pi*a*εo) * [- 6/4 + 6/4√3 - 3/8]

= (q^2)/(pi*a*εo) * [√3/2-15/8].

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