Two small spheres, each with mass m=3g and charge q, are suspended from a point by threads of length L=0.22m. What is the charge on each sphere if the threads make an angle theta of 15 degrees with the vertical?
To find the charge on each sphere, we can use the concept of electrostatic equilibrium. In electrostatic equilibrium, the net force and net torque acting on a charged object are both zero.
Let's break down the problem step by step:
Step 1: Calculate the weight of each sphere
The weight of each sphere can be calculated using the formula:
Weight = mass × gravity
Given that the mass of each sphere is 3g (grams) and the acceleration due to gravity is approximately 9.8 m/s², we can convert the mass to kilograms and calculate the weight:
Mass = 3g = 3 × 0.001 kg (1 gram = 0.001 kg)
Weight = 3 × 0.001 kg × 9.8 m/s²
Step 2: Decompose the weight vector into horizontal and vertical components
Since the threads are at an angle with the vertical, we need to decompose the weight vector into horizontal and vertical components.
The vertical component can be calculated using the formula:
Vertical Component = Weight × cos(theta)
And the horizontal component can be calculated using the formula:
Horizontal Component = Weight × sin(theta)
Where theta is the angle the threads make with the vertical, which is 15 degrees in this case.
Step 3: Equate the electrostatic and gravitational forces
In electrostatic equilibrium, the force due to the electrical repulsion of the spheres must balance the tension in the threads. Since the threads are assumed to be massless, the tension is equal to the horizontal component of the weight:
Tension = Horizontal Component
Step 4: Calculate the electrostatic force between the spheres
The electrostatic force between the spheres can be calculated using Coulomb's law formula:
Electrostatic Force = k × (q^2 / distance^2)
Where k is the electrostatic constant and q is the charge on each sphere. Since both spheres have the same charge, we can replace q^2 with q1 × q2.
Step 5: Find the distance between the spheres
The distance between the spheres is given by the length of the threads, which is 0.22m.
Step 6: Equate the electrostatic force and tension
Since the net force should be zero in equilibrium, we can equate the electrostatic force to the tension:
k × (q^2 / distance^2) = Tension
Step 7: Solve for the charge on each sphere
Simplifying the equation, we have:
q^2 = (Tension × distance^2) / k
Finally, we can solve for q by taking the square root of both sides of the equation:
q = √((Tension × distance^2) / k)
Given all the provided values, you can plug them into the equation and calculate the charge on each sphere.