A small plastic ball of mass m = 3.00 g is suspended by a string of length L = 19.5 cm in a uniform electric field, as shown in the figure below. If the ball is in equilibrium when the string makes a θ = 17.0° angle with the vertical as indicated, what is the net charge on the ball?
I assume that E is horizontal
look at tension T in string
force up = mg
force sideways = k Q E
string does that with tension
so
T cos 17 = mg
T sin 17 = k QE
sin/cos = tan
so
tan 17 = k Q E/mg
To find the net charge on the ball, we can use the principle of electrostatic equilibrium. In this situation, the gravitational force acting on the ball is balanced by the electrostatic force due to the electric field.
First, let's express the gravitational force acting on the ball. The weight (W) of the ball can be calculated using the formula:
W = mg
where m is the mass of the ball and g is the acceleration due to gravity. In this case, m = 3.00 g and g = 9.8 m/s^2.
W = (3.00 g) x (9.8 m/s^2)
W = 29.4 g m/s^2
Next, we need to determine the electrostatic force acting on the ball. The electrostatic force (Fe) experienced by a charged object in an electric field (E) is given by the formula:
Fe = qE
where q is the charge on the object and E is the electric field strength. In this situation, the electric field is uniform, so the magnitude of the electric field (E) is the same as the magnitude of the gravitational field strength (g). Therefore, E = g = 9.8 m/s^2.
Since the ball is in equilibrium, the electrostatic force (Fe) must have the same magnitude as the gravitational force (W) but act in the opposite direction. Therefore, we can equate these two forces:
Fe = -W
Substituting the formulas for Fe and W, and considering that Fe = qE, we have:
qE = -W
Plugging in the values we already obtained:
q(9.8 m/s^2) = -(29.4 g m/s^2)
To solve for the charge (q), we divide both sides of the equation by 9.8 m/s^2:
q = -(29.4 g m/s^2) / (9.8 m/s^2)
q = -3.00 g
Finally, we can convert the charge from grams to Coulombs using the conversion factor:
1 g = 1 x 10^-3 kg
1 C = 1 A.s
Thus, the net charge on the ball is:
q = -3.00 g x (1 x 10^-3 kg/g)
q = -3.00 x 10^-3 kg
Please note that the negative sign indicates that the ball has a negative charge. Since the net charge is given in kilograms, it is a very small charge, which is typical for charged objects in electrostatic experiments.