Find the Charge on the Spheres Two identical small charged spheres, each having a mass of 7.0 x 10-2 kg, hang in equilibrium as shown in figure (a). The length L of each string is 0.100 m, and the angle 8 is 5.00. Find the magnitude of the charge on each sphere. (a) Two identical spheres, each carrying the (b) Diagram of the forces acting on same charge q, suspended in equilibrium. the sphere on the left. SOLUTION O mg O 9 + OF electric force from the other sphere mg gravitational force O 07 L mg Conceptualize Figure (a) helps us conceptualize this example. The two spheres exert ---Select--forces on each other. If they are held close to each other and released, they move outward from the center and settle into the configuration in figure (a) after the oscillations have vanished due to air resistance. Categorize The key phrase "in equilibrium" helps us model each sphere as a particle --Select--- force. (3) tan(8) = - Hint L Analyze The force diagram for the left-hand sphere shown in figure (b). The sphere is in equilibrium under the application of the following forces: force from the string O (4) sin(0) = a= + |g| = 9 = 2q T cos 0 + e 0 mg (For the next two answers, use the following as necessary: 8, m, g, and F. Do not substitute numerical values; use variables only.) From the particle in equilibrium model, set the net force on the left-hand sphere equal to zero for each component: Use the geometry of the right triangle in figure (a) to find a relationship between a, L, and 8: 0 (1) Fx = T sin(8) - F = 0 sin(0) = (2) Fy = T cos(0) - mg = 0 T cos(8) = (For the next three answers, use the following as necessary: 8, m, g, Fe, L, and ke. Do not substitute numerical values; use variables only.) Divide Equation (1) by Equation (2) to find F: Tsin 0 q Solve Coulomb's law for the charge lgl on each sphere and substitute from Equations (3) and (4): F.(2a) Fr Ke Substitute numerical values (Give your answer in C.): |g| = Finalize If the sign of the charges were not given in figure (a), we could not determine them. In fact, the sign of the charge is not important. The situation is the same whether both spheres are positively charged or negatively charged. EXERCISE Which figure below most accurately describes the forces acting between the two positive charges 29 and q? v This example is similar to the particle in equilibrium problems in an earlier chapter with the added feature that one of the forces on a sphere is an electric 2q q 2q q