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.
We can start by using Coulomb's Law to find the force of repulsion between the two charges:
F = (k*q1*q2)/r^2
where k is Coulomb's constant (9.0x10^9 N*m^2/C^2), q1 and q2 are the magnitudes of the charges (-3µC and -4µC), and r is the distance between them.
To find the distance r, we can use trigonometry and the fact that the two spheres are hanging in equilibrium, which means that the force of repulsion is balanced by the force of gravity.
The force of gravity on each sphere is:
Fg = mg
where m is the mass of the sphere and g is the acceleration due to gravity (9.8 m/s^2).
Using trigonometry, we can find the horizontal and vertical components of the force of gravity:
Fgx = Fg*sin(θ) = (5.0x10^2 kg)*(9.8 m/s^2)*sin(15°) = 129 N
Fgy = Fg*cos(θ) = (5.0x10^2 kg)*(9.8 m/s^2)*cos(15°) = 475 N
Since the spheres are in equilibrium, the horizontal component of the force of repulsion (Fr*cos(15°)) must be equal to Fgx, and the vertical component of the force of repulsion (Fr*sin(15°)) must be equal to Fgy.
So we have two equations:
Fr*cos(15°) = 129 N
Fr*sin(15°) = 475 N
Solving for Fr in the first equation and substituting into the second equation, we get:
Fr = 129 N / cos(15°) = 134 N
134 N * sin(15°) = (9.0x10^9 N*m^2/C^2)*(-3µC)*(-4µC)/r^2
Simplifying the right side, we get:
2.16x10^-5 = (12 µC^2)/(r^2)
Solving for r, we get:
r = √(12 µC^2 / 2.16x10^-5) ≈ 1.12 m
So the distance that separates the charges is approximately 1.12 meters.
To find the distance that separates the charges, we can use Coulomb's law and the equilibrium condition.
Coulomb's law states that the force between two point charges is given by:
F = (k * q1 * q2) / r^2
where F is the force, k is the electrostatic constant (9.0 x 10^9 N*m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Since the spheres are in equilibrium, the electrostatic force between them is equal to the gravitational force acting on each sphere.
The electrostatic force between the spheres is given by Coulomb's law:
Fe = (k * q1 * q2) / r^2
The gravitational force acting on each sphere is given by:
Fg = m * g
where m is the mass of the sphere and g is the acceleration due to gravity (9.8 m/s^2).
In equilibrium, both forces are equal, so we can set them equal to each other:
Fe = Fg
(k * q1 * q2) / r^2 = m * g
Substituting the given values, we have:
(9.0 x 10^9 N*m^2/C^2) * (-3 x 10^-6 C) * (-4 x 10^-6 C) / r^2 = (5.0 x 10^-2 kg) * (9.8 m/s^2)
Simplifying this equation, we have:
(108 x 10^3 N*m^2)/(C^2) / r^2 = 0.49 N
Multiplying both sides of the equation by r^2, we get:
(108 x 10^3 N*m^2)/C^2 = 0.49 N * r^2
Dividing both sides of the equation by (0.49 N), we get:
r^2 = (108 x 10^3 N*m^2)/(0.49 N)
r^2 = 2.2 x 10^5 m^2
Taking the square root of both sides, we get:
r = √(2.2 x 10^5 m^2)
r ≈ 470 m
Therefore, the distance that separates the charges is approximately 470 meters.