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Two identical conducting spheres (of negligible radius), having charges of opposite sign, attract each other with a force of 0.108 N when separated by 0.5 meter. 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 0.036 N keeping the distance same. What were the initial charges on the spheres?
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Two identical conducting spheres (of negligible radius), having charge...
Initial setup:
- Two identical conducting spheres with negligible radius
- Charges of opposite sign
- Attract each other with a force of 0.108 N
- Separated by 0.5 meter

After connecting and disconnecting:
- Connected by a conducting wire and then removed
- Repel each other with a force of 0.036 N
- Distance between spheres remains 0.5 meter

To find: Initial charges on the spheres

Solution:
Step 1: Calculate the initial force between the spheres
- Using Coulomb's Law: F = (k*q1*q2)/r^2, where k is Coulomb's constant, q1 and q2 are charges on the spheres, and r is the distance between them
- Rearranging the equation: q1*q2 = (F*r^2)/k
- Substituting values: q1*q2 = (0.108 N)*(0.5 m)^2/(9*10^9 N m^2/C^2) = 1.2*10^-9 C^2

Step 2: Calculate the final charges on the spheres
- After connecting the spheres, the charges will redistribute themselves to reach equilibrium
- Since the spheres are identical, they will have equal charges after redistribution
- Let the final charge on each sphere be qf
- Using the principle of conservation of charge: q1 + q2 = 2*qf
- Substituting q1*q2 from step 1: (qf)^2 = (1.2*10^-9 C^2)/4 = 0.3*10^-9 C^2
- Taking the square root: qf = 0.547 nC (nanoCoulombs)

Step 3: Calculate the initial charges on the spheres
- Let the initial charge on one sphere be qi
- The other sphere will have charge -qi (opposite sign)
- Using the principle of conservation of charge: qi - qi = 2*qf
- Substituting values: qi = qf = 0.547 nC

Final answer: The initial charges on the spheres were 0.547 nC, with one sphere having charge +0.547 nC and the other having charge -0.547 nC.
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Two identical conducting spheres (of negligible radius), having charges of opposite sign, attract each other with a force of 0.108 N when separated by 0.5 meter. 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 0.036 N keeping the distance same. What were the initial charges on the spheres?
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