two similar charged small spheres each having a charge of 0.044 micro coulombs are suspended from a point by a thread of length 15 cm each. at equilibrium the angle is 5 degree. what is the mass of each sphere ?
Tsin(α/2)=kq²/{2Lsin(α/2)}²,
Tcos(α/2)=mg.
tan(α/2)= kq²/mg{2Lsin(α/2)}²,
m= kq²/g{2Lsin(α/2)}²•tan(α/2)=...
To find the mass of each sphere, we need to use the principle of electrostatic equilibrium. This principle states that the electrical force between two charged objects is equal to the force of gravity acting on each object.
First, let's calculate the force of gravity acting on each sphere. We know that the spheres are suspended from a point by a thread, so the force of gravity acting on each sphere is given by:
F_gravity = m * g
Where:
F_gravity is the force of gravity
m is the mass of each sphere
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, let's calculate the electrical force between the spheres. The electrical force between two charged objects is given by Coulomb's Law:
F_electrical = (k * q1 * q2) / r^2
Where:
F_electrical is the electrical force
k is the electrostatic constant (approximately 9 x 10^9 Nm^2/C^2)
q1 and q2 are the charges of the two spheres (0.044 microCoulombs each)
r is the distance between the spheres (twice the length of the thread, or 2 * 15 cm = 30 cm)
Given that the spheres are in electrostatic equilibrium, the electrical force between them must be equal to the force of gravity on each sphere. Therefore, we can equate these two forces:
F_gravity = F_electrical
m * g = (k * q1 * q2) / r^2
Now, let's plug in the values and calculate the mass of each sphere.
m * 9.8 = (9 * 10^9 * 0.044 * 10^-6 * 0.044 * 10^-6) / (0.30)^2
Simplifying the equation:
m = ((9 * 10^9 * 0.044 * 10^-6 * 0.044 * 10^-6) / (0.30)^2) / 9.8
m ≈ 1.29 * 10^-7 kg
So, the mass of each sphere is approximately 1.29 * 10^-7 kg.