A small 6.8 g plastic ball is suspended by a
37.3 cm long string in a uniform electric field
of 4790 N/C, as shown.
The acceleration of gravity is 9.8 m/s2 , and
the Coulomb constant is 8.99×109 N · m2/C2..
If the ball is in equilibrium when the string
makes a 11.2◦ angle with the vertical as indicated, what is the net charge on the ball?
Answer in units of μC.
To find the net charge on the ball, we need to use the principles of electrostatics and equilibrium.
First, we need to understand the forces acting on the ball. In this case, there are two forces at play: the gravitational force (Fg) and the electrostatic force (Fe).
The gravitational force can be calculated using the formula:
Fg = m * g
where m is the mass of the ball and g is the acceleration due to gravity.
Given that the mass of the ball is 6.8 g, and the acceleration due to gravity is 9.8 m/s^2, we can calculate the gravitational force:
Fg = (6.8 g) * (9.8 m/s^2)
Next, we need to determine the electrostatic force acting on the ball. The electrostatic force is given by Coulomb's Law:
Fe = k * (q1 * q2) / r^2
where k is the Coulomb constant, q1 and q2 are the charges on the two objects, and r is the distance between them.
In this case, the ball is in equilibrium, which means the electrostatic force must balance the gravitational force. Therefore, Fe = Fg.
We can rearrange Coulomb's Law equation to solve for the net charge on the ball (q):
q = (Fe * r^2) / (k * m)
Using the given values for r (37.3 cm or 0.373 m), k (8.99 × 10^9 N · m^2/C^2), and m (6.8 g or 0.0068 kg), we can substitute these into the equation to find the net charge on the ball.
q = ((Fg * r^2) / m) / k
Finally, we can substitute the calculated values into the equation and solve for q:
q = ((Fg * r^2) / m) / k = ((m * g * r^2) / m) / k
You can now substitute the given values for mass (m), acceleration due to gravity (g), distance (r), and Coulomb constant (k) into the equation and calculate the net charge on the ball.