The third charge q 3 must lie between the other two or else the forces acting (a) If the system of three charges is to be in equilibrium, the force on each charge.Note that we may easily verify that the force on 4 q also The signs are chosen so that a negative force value would cause q to move leftward. # + from which we obtain Q = 55 μC 55 C≈ μ. (b) Now, we require that the x components cancel, and we note that in this case, the angle of force on q 3 exerted by Q is +θ (it is repulsive, and Q is positive-valued). What if we had not made the assumption, above, that q 1 + q 2 ≥ 0? If the signs of theĬharges were reversed (so q 1 + q 2 0) we conclude Q = –83 μC. Originally had charge –1 × 10 –6 C and the other had charge +3 × 10 –6 C. Note that since the spheres are identical, the solutions are essentially the same: one sphere (b) If we instead work with the q 1 = –1 × 10 –6 C root, then we find q 2 =×3 10 C− 6. If the positive sign is used, q 1 = 3 × 10 –6 C, and if the negative sign is used, Multiplying by q 1 and rearranging, we obtain a quadratic equation qq 126 −× −× =c2 00 10.−−CCh 1 3 00 10 1220 Which we substitute into the sum result, producing Where we have taken the positive root (which amounts to assuming q 1 + q 2 ≥ 0). We solve the two force equations simultaneously for q 1 and q 2. The force is now one of repulsion and is given by This means the charge onĮach sphere is ( q 1 + q 2 )/2. Is conserved, the total charge is the same as it was originally. The wire is connected, the spheres, being identical, acquire the same charge. The negative sign indicates that the spheres attract each other. We choose the coordinate system so the force on q 2 is positive if it is repelled by Spherically symmetric and Coulomb’s law can be used. Then the charge distribution on each of them is The magnitude of the force of either of the charges on the other is given by Setting the derivative dF dq / equal to zero leads to Q – 2 q = 0, We want the value of q that maximizes theįunction f ( q ) = q ( Q – q ). Where r is the distance between the charges. The repulsive force between spheres 1 and 2 is finally Therefore, the charge of sphereģ is 3 q /4 in the final situation. Then sphere 3 (now carrying charge q /2) is brought into contact with sphere 2, a totalĪmount of q /2 + q becomes shared equally between them. When the neutral sphere 3 touches sphere 1, sphere 1’s charge decreases to
Spheres 1 and 2 each having charge q and experiencing a mutual repulsive forceįkqr = 22 /. Uncharged one, they will (fairly quickly) each attain half that charge ( q /2). Therefore, when a charged sphere ( q ) touches an
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