Question

Two identical spheres of negligible radius having charges of opposite sign attract each other with a force of 0.108N when separated by 0.5m. The spheres are connected by a conducting wire which is then removed (when charge stops flowing) and thereafter repel each other with a force of 0.036N keeping the distance same. What were the initial charges on the spheres?

Answer 1

here,
F1 = 0.108 N
r = 0.5 m
F2 = 0.036 N
let charges on the spheres be q1 and q2 respectively.
charges after connection gets redistributed in the spheres , so let charge on each sphere be q.
from conservation of charge
q1 + q2 = 2q...........eq.1

again
F1 = k q1q2/r²...........eq.2
F2 = k q²/r².............eq.3

=> 0.036 = 9×10⁹ ×q²/(0.5)²
=> q² = 0.036 × (0.5)²/9×10⁹
=> q² = 1×10⁻¹²
=> q=1×10⁻⁶ C.............eq.4

from eq 2 and 3
F1/F2 = q1q2/q² 
=> 3 = q1q2/q²
3q² = q1q2
=> q1q2 = 3q²

again,

(q1 ₋ q2)² = (q1 + q2)² ₋ 4q1q2
 = (2q)² ₋ 3q²/r²
 = q²/r²

by eq.4
(q1₋q2) = q/r
=1×10⁻⁶/0.5
=2×10⁻⁶..............eq.5

from eq.1 and eq.4
q1 + q2 = 4×10⁻⁶..............eq.6

adding 5 and 6
q1 = 3×10⁻⁶
q1 = 3μC

subtracting 5 from 6
q2 = 1×10⁻⁶
q2 = 1μC

Answer 2

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