Two identical, small, charged spheres, each having a mass of `4.0xx10^(-2)`kg hang in equilibrium as shown in Fig. 22-10a. The length of each string is 1.5m and the angle `theta` is `37.0^(@)`. Find the magnitude of the charge on each sphere.

Explanation

KEY IDEA
From Fig. 22-10a, we can see that 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 Fig. 22-10a after the damped oscillations due to air resistance have vanished.
The key phrase "in equilibrium" helps us to classify this as an equilibrium problem, which we approach as we did equilibrium problems in the chapter on Newton.s Laws in (Volume 1) with the added feature that one of the forces on a sphere is an electric force. We analyze this problem by drawing the free-body diagram for the left-hand sphere in Fig. 22-10b. The sphere is in equilibrium under the application of forces T from the string, the electric force `F_(e )` from the other sphere, and the gravitational force mg.
Calculations: Because the sphere is in equilibrium, the forces in the horizontal and vertical directions must separately add up to zero.
`sum F_(x)=T sin theta-F_(e ) " " (22-11)`
`sum F_(y)=T cos theta-mg " " (22-12)`
Frow Eq. 22-12, we see that `T=mg//cos theta`, thus, T can be eliminated from Eq. 22-11 if we make this substitution. This gives a value for the magnitude of the electric force `F_(e ): `
`F_(e )=mg tan theta =(4xx10^(-2)kg) (10 m//s^(2)) tan 37^(@)=0.3N`.
Considering the geometry of the triangle in Fig. 22-10a, we see that `sin theta=alpha//L`. Therefore,
`a=L sin theta=(1.5m)sin37^(@)=0.9m`
The separation of the spheres is `2a=1.8m`. From Coulomb.s law (Eq. 22-10), the magnitude of the electric froce is
`F_(e )=k_(e ) (|q|^(2))/(r^(2))`
where r = 2a = 1.8m and |q| is the magnitude of the charge on each sphere. This equation can be solved for `|q|^(2)` to give
`|q|^(2)=(F_(3)r^(2))/(k_(e ))=(0.3xx(1.8)^(2))/(9xx10^(9)Nm^(2)//C^(2))=1.08xx10^(-10)C^(2)`
`|q|=6sqrt(3)muC`