Question
Two small charged sphere each with magnitude of 3µC and have a masses of 5.0x10^2kg .Hang in equilibrium.The length of each string is 155mm.Find the angle between the sphere.
Answers
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We can start by finding the gravitational force and the electrostatic force acting on each sphere.
The gravitational force is given by:
Fg = mg
where m is the mass of the sphere and g is the acceleration due to gravity (9.81 m/s^2). So for each sphere:
Fg = (5.0x10^-3 kg)(9.81 m/s^2) = 4.91x10^-2 N
The electrostatic force between two charged spheres is given by Coulomb's law:
Fe = kq1q2/r^2
where k is the Coulomb constant (9x10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges on the spheres, and r is the distance between the centers of the spheres. Since the spheres are hanging in equilibrium, the electrostatic force must be equal and opposite to the gravitational force, so:
kq^2/r^2 = mg
Solving for q, we get:
q = sqrt(mgr^2/k)
Plugging in the values, we get:
q = sqrt((5.0x10^-3 kg)(9.81 m/s^2)(0.155 m)^2/(9x10^9 N m^2/C^2)) = 3.26x10^-6 C
Now we can use trigonometry to find the angle between the spheres. Let's call this angle θ. The electrostatic force between the spheres is the tension in the strings, so:
Fe = T = 2mg sin(θ/2)
The factor of 2 comes from the fact that each sphere is connected to the ceiling by its own string. Solving for θ, we get:
θ = 2 sin^-1(T/2mg)
Plugging in the values, we get:
θ = 2 sin^-1((4.91x10^-2 N)/(2(5.0x10^-3 kg)(9.81 m/s^2))) = 0.335 radians
Converting to degrees, we get:
θ = 19.2 degrees
So the angle between the spheres is 19.2 degrees.
The gravitational force is given by:
Fg = mg
where m is the mass of the sphere and g is the acceleration due to gravity (9.81 m/s^2). So for each sphere:
Fg = (5.0x10^-3 kg)(9.81 m/s^2) = 4.91x10^-2 N
The electrostatic force between two charged spheres is given by Coulomb's law:
Fe = kq1q2/r^2
where k is the Coulomb constant (9x10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges on the spheres, and r is the distance between the centers of the spheres. Since the spheres are hanging in equilibrium, the electrostatic force must be equal and opposite to the gravitational force, so:
kq^2/r^2 = mg
Solving for q, we get:
q = sqrt(mgr^2/k)
Plugging in the values, we get:
q = sqrt((5.0x10^-3 kg)(9.81 m/s^2)(0.155 m)^2/(9x10^9 N m^2/C^2)) = 3.26x10^-6 C
Now we can use trigonometry to find the angle between the spheres. Let's call this angle θ. The electrostatic force between the spheres is the tension in the strings, so:
Fe = T = 2mg sin(θ/2)
The factor of 2 comes from the fact that each sphere is connected to the ceiling by its own string. Solving for θ, we get:
θ = 2 sin^-1(T/2mg)
Plugging in the values, we get:
θ = 2 sin^-1((4.91x10^-2 N)/(2(5.0x10^-3 kg)(9.81 m/s^2))) = 0.335 radians
Converting to degrees, we get:
θ = 19.2 degrees
So the angle between the spheres is 19.2 degrees.