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.
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.
To find the angle between the spheres, we need to analyze the forces acting on each sphere.
1. Gravity Force:
The force due to gravity acting on each sphere is given by:
F_gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity.
For each sphere:
F_gravity = (5.0x10^(-2) kg) * (9.8 m/s^2)
F_gravity = 0.49 N
2. Electrostatic Force:
The electrostatic force between the two spheres is given by Coulomb's law:
F_electrostatic = (k * |Q1 * Q2|) / r^2
where k is the electrostatic constant (8.99x10^9 N m^2/C^2), Q1 and Q2 are the charges on the spheres, and r is the distance between the spheres.
For each sphere:
F_electrostatic = (8.99x10^9 N m^2/C^2) * (3x10^(-6) C) * (3x10^(-6) C) / (0.155 m)^2
F_electrostatic = 3.29 N
3. Tension in the Strings:
The tension in the strings balances the combined forces of gravity and electrostatic force.
Since the spheres are in equilibrium, the tension in each string must be equal to the forces acting on each sphere:
T = F_gravity + F_electrostatic
T = 0.49 N + 3.29 N
T = 3.78 N
Now, let's find the angle between the spheres.
Consider a triangle formed by the two strings and the line connecting the spheres.
Let's call the angle between the line connecting the spheres and one of the strings θ.
Using trigonometry, we can relate the tension in the strings T with the angles involved:
T / sin(θ) = T / sin(θ) = T / sin(180° - θ)
Since tension T is the same for both strings, we can write:
sin(θ) = sin(180° - θ)
To solve for θ, we can take the inverse sine of both sides:
θ = sin^(-1)(sin(180° - θ))
Calculating this using a scientific calculator or software, we find:
θ ≈ 52.81°
Therefore, the angle between the spheres is approximately 52.81 degrees.