Two small charged spheres repel each other with a force of
2 × 10^−3 N. The charge on one sphere is twice that on the other. When
one of the charges is moved 10 cm away from the other, the force is
5 × 10^−4 N. Calculate the charges and the initial distance between
them.

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2014-12-26T02:43:35+05:30

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Distance between the charges can be obtained simply by, noticing that force becomes 1/4 when distance is increased. It means distance becomes double.  Hence, 10cm becomes 20 cm.  So d = 10cm.
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   electrostatic force F =  K Q * 2Q / d²
         Q = charge on one sphere,  2 Q = charge on the other
           d= distance between them
         K = 9 * 10⁹ N-m²/C²

     2 * 10⁻³ = 2 K Q² / d²  => K Q² = 10⁻³ d²
 
   forces are moved away 10 cm = 0.10 meter.
  
     5 * 10⁻⁴ = 2 K Q² / (d+0.10)²
        substitute the value of KQ² from earlier equation,
         5 * 10⁻⁴ (d+0.10)² = 2 * 10⁻³ d²
         d + 0.10 = 2 d
============================
  so           d = 0.10 meter = 10 cm
   SO,         Q² = 10⁻³ * 0.10² / (9 * 10⁹)
               Q = 0.03 micro coulomb or  30 nano Coulombs

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