Example 23.4 Find the Charge on the Spheres Problem Two identical small charged

Question

Example 23.4 Find the Charge on the Spheres Problem Two identical small charged spheres, each having a mass of 2.0 10-2 kg, hang in equilibrium as shown in Figure 23.10a. The length of each string is 0.05 m, and the angle is 5.0°. Find the magnitude of the charge on each sphere. Figure 23.10a
Two identical spheres, each carrying the same charge q, suspended in equilibrium. Solution Figure 23.10a helps us conceptualize this problem - the two spheres exert repulsive forces on each other. If they are held close to each other and released, they will move outward from the center and settle into the configuration in Figure 23.10a after the damped oscillations due to air resistance have vanished. The key phrase "in equilibrium" helps us categorize this as an equilibrium problem, which we approach as we did equilibrium problems in Chapter 5 with the added feature that one of the forces on a sphere is an electric force. We analyze this problem by drawing a free-body diagram for the left-handed sphere in Figure 23.10b. The sphere is in equilibrium under the application of the forces T from the string, the electric force Fe from the other sphere, and the gravitational force mg. Figure 23.10b. The free body diagram for the left sphere. Because the sphere is in equilibrium, the forces in the horizontal and vertical directions must separately add up to zero: (1) (2) From Equation (2), we see that T = mg/cos ; thus, T can be eliminated from Equation (1) if we make this substitution. This gives a value for the magnitude of the electric force Fe: N Considering the geometry of the right triangle in Figure 23.10a, we see that sin = a/L. Therefore, m The separation of the sphere is 2a.

From Coulomb's law (Eq. 23.1), the magnitude of the electric force is

where r = 2a and |q| is the magnitude of the charge on each sphere. (Note that the term |q|2 arises here because the charge is the same on both spheres.) This equation can be solved for |q|2 to give C2 C To finalize the problem, note that we found only the magnitude of the charge |q| on the spheres. There is no way we could find the sign of the charge from the information give. In fact, the sign of the charge is not important. The situation will be exactly the same whether both spheres are positively charged or negatively charged.What If? Suppose your roommate proposes solving this problem without the assumption that the charges are of equal magnitude. She claims that the symmetry of the problem is destroyed if the charges are not equal, so that the strings would make two different angles with the vertical, and the problem would be much more complicated. How would you respond?

Answer. You should argue that the symmetry is not destroyed and the angles remain the same. Newton's third law requires that the electric forces on the two charges be the same, regardless of the equality or nonequality of the charges. The solution to the example remains the same through the calculation of |q|2. In this situation, the value corresponds to the product q1q2, where q1 and q2 are the values of the charges on the two spheres. The symmetry of the problem would be destroyed if the masses of the spheres were not the same. In this case, the strings would make different angles with the vertical and the problem would be more complicated.

Explanation / Answer

Fe= mg tan5

Fe=(2x10-2)(9.8)tan5

Fe=0.01715N

a=Lsin5=0.05sin5=0.00436

r=2a=2(0.00436)=0.00872m

Fe= Kq2/r2

q2=Fe r2/K

q2=(0.01715)(0.00872)2/(9x109)

q=1.2x10-8C

The forces are equal in magnitude but opposite in direction . so option (b) is correct.

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