Two identical small charged spheres, each having a mass of 5.0x102 kg and having the
magnitudes of -3µC and -4µC, hang in equilibrium. The angle θ is
15.0°. Find distance that separates the charge.( Important constants: Charge 8.99x109
, acceleration due gravity 9.8m/s2).
We can use Coulomb's law to find the force between the two charges:
F = k * q1 * q2 / r^2
where k is Coulomb's constant (8.99x10^9), q1 and q2 are the magnitudes of the charges (-3µC and -4µC), and r is the distance between them.
Since the spheres are in equilibrium, the net force on each sphere must be zero. We can draw a free body diagram for one of the spheres:
- There is a downward gravitational force of mg (where m is the mass of the sphere and g is the acceleration due to gravity)
- There is an upward electrostatic force from the other sphere
The angle θ is given in the diagram, so we can split the electrostatic force into horizontal and vertical components:
F_horizontal = F * sin(θ)
F_vertical = F * cos(θ)
Since the sphere is in equilibrium, the vertical forces must balance:
F_vertical = mg
We can solve for F using this equation, and then use the horizontal component to find the distance between the charges:
F = mg / cos(θ)
r = √(k * q1 * q2 / F)
Plugging in the given values:
F = (5.0x10^2 kg) * (9.8 m/s^2) / cos(15°) ≈ 2486 N
r = √(8.99x10^9 N*m^2/C^2 * (-3x10^-6 C) * (-4x10^-6 C) / 2486 N) ≈ 0.0186 m
Therefore, the distance between the charges is approximately 0.0186 meters.
To solve this problem, we can use Coulomb's Law and the gravitational force equation.
The Coulomb's Law equation is: F = k * (q1 * q2) / r^2, where F is the electrostatic force, k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
The gravitational force equation is: F = m * g, where F is the gravitational force, m is the mass, and g is the acceleration due to gravity (9.8 m/s^2).
Since the two charges are in equilibrium, the electrostatic force and the gravitational force are equal:
k * (q1 * q2) / r^2 = m * g
First, let's find the gravitational force for each charge:
Fg1 = m * g = (5.0 x 10^2 kg) * (9.8 m/s^2) = 4900 N
Fg2 = m * g = (5.0 x 10^2 kg) * (9.8 m/s^2) = 4900 N
Now, solve for the distance between the charges (r):
k * (q1 * q2) / r^2 = m * g
k * (-3 x 10^-6 C) * (-4 x 10^-6 C) / r^2 = 4900 N
8.99 x 10^9 Nm^2/C^2 * 12 x 10^-12 C^2 / r^2 = 4900 N
(8.99 x 10^9 Nm^2/C^2 * 12 x 10^-12 C^2) / (4900 N) = r^2
(8.99 x 10^9 Nm^2/C^2 * 12 x 10^-12 C^2) / (4900 N) = r^2
r^2 = 2.19 x 10^-6 m^2
r = √(2.19 x 10^-6) m
r ≈ 1.48 x 10^-3 m
Therefore, the distance separating the charges is approximately 1.48 x 10^-3 meters.