A positively charged particle Q1 = +45 nC is held fixed at the origin. A second charge Q2 of mass m = 7.5 μg is floating a distance d = 25 cm above charge Q1. The net force on Q2 is equal to zero. You may assume this system is close to the surface of the Earth.
Calculate the magnitude of Q2 in units of nanocoulombs.
To calculate the magnitude of Q2 in units of nanocoulombs, we need to find the value of Q2 when the net force on it is equal to zero.
The net force on Q2 is determined by the electrical force and the force due to gravity. The electrical force between two charged particles is given by Coulomb's Law:
F_electric = (k * |Q1 * Q2|) / r^2
Where:
- F_electric is the electrical force between the charges
- k is the electrostatic constant with a value of approximately 9 x 10^9 Nm^2/C^2
- |Q1| and |Q2| are the magnitudes of the charges
- r is the distance between the charges
The force due to gravity is given by:
F_gravity = m * g
Where:
- F_gravity is the force due to gravity
- m is the mass of Q2
- g is the acceleration due to gravity, approximately 9.8 m/s^2 near the surface of the Earth
Since the net force on Q2 is zero, we can set the electrical force equal to the force due to gravity:
(k * |Q1 * Q2|) / r^2 = m * g
Rearranging this equation to solve for Q2:
|Q2| = (m * g * r^2) / (k * |Q1|)
Now we can substitute the given values into the equation:
m = 7.5 μg = 7.5 x 10^-9 kg
g = 9.8 m/s^2
r = 25 cm = 0.25 m
k = 9 x 10^9 Nm^2/C^2
|Q1| = +45 nC = 45 x 10^-9 C
Substituting these values:
|Q2| = (7.5 x 10^-9 kg * 9.8 m/s^2 * (0.25 m)^2) / (9 x 10^9 Nm^2/C^2 * 45 x 10^-9 C)
Calculating this expression will give us the magnitude of Q2 in units of nanocoulombs.