OBSERVATION: Two charged conducting spheres hanging from a pivot repel each othe
ID: 1652572 • Letter: O
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OBSERVATION: Two charged conducting spheres hanging from a pivot repel each other CONJECTURE: If a plastic rod is rubbed with fur the rod will become negatively charged. If a conducting sphere comes into contact with the rod the sphere will become negatively charged. If another identical sphere comes into contact with the charged sphere the charge will be equally distributed between the two spheres. If both spheres are attached to string of equal length and are hung from the same pivot the spheres will repel each other as shown in Figure 1. If the mass of the spheres, the separation distance, and the length of the string are known the charge can be determined. Figure 1: Identically charged spheres repel EXPERIMENT: For the following assume the conducting spheres are point-like, are of equal mass, and share charge equally following contact (this is not a bad assumption due to the symmetry). (1) Consider Figure 1 above and derive an expression relating the charge to the observables. (Hint: draw a free-body diagram for one of the charges and apply Newton's Laws of Motion, and apply the small-angle approximation tan sin .)Explanation / Answer
Lets examine the forces on left charge,
The electrostatic force due to repulsion acts leftward.
The weight acts downwards.
The tension T acts at an angle phi with vertical which balances both weight and electrostatic repulsion.
So T cos phi = mg
T sin phi = kq^2 /(2L sin phi)^2
Dividing second equation by first,
tan phi = kq^2 /[(2L sin phi)^2 *mg]
q^2 = [(2L sin phi)^2 *(mg tan phi)/k ]
q = sqrt ( [(2L sin phi)^2 *(mg tan phi)/k ])
After small angle approximation,
q = [(4L^2 *mg (sin phi)^3 /k ]
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