Two 0.20 g metal spheres are hung from a common point by non-conducting threads that are 30 cm long. Both are given identical charges, and the electrostatic repulsion forces them apart until the angle between the threads is 20 degrees. How much charge was placed on each sphere?
20nC
4t
To find the charge on each sphere, we can use Coulomb's Law and the fact that the electrostatic repulsion force is balanced by the tension in the threads.
The formula for Coulomb's Law is:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the Coulomb's constant (approximately 9 × 10^9 N m^2/C^2)
q1 and q2 are the charges on the spheres
r is the distance between the centers of the spheres
In this case, the electrostatic force is balanced by the tension in the threads, so we can set the two forces equal to each other:
Tension = F
The electrostatic force can also be written as:
F = k * (q1 * q2) / (0.3^2) (since the distance between the centers of the spheres is equal to twice the length of the thread)
Now, we can substitute this value of F into the equation for tension:
Tension = k * (q1 * q2) / (0.3^2)
Since both spheres have identical charges, we can simplify the equation to:
Tension = k * (q^2) / (0.3^2)
where q is the charge on each sphere.
We also know that the angle between the threads is 20 degrees. The force components acting along the threads are equal to the tension in the threads. This can be written as:
Tension * sin(θ) = mg
Where:
m is the mass of the spheres
g is the acceleration due to gravity
θ is the angle between the threads (20 degrees or 0.35 radians)
The mass of each sphere can be found using their weight:
m = Weight / g
Since both spheres have the same mass, we can write:
2 * (Weight / g) = m
Combining these equations, we have:
Tension * sin(θ) = 2 * (Weight / g)
Substituting the value for the tension we found earlier:
k * (q^2) / (0.3^2) * sin(θ) = 2 * (Weight / g)
Now, we need to find the weight of each sphere:
Weight = mass * g
Substituting this value into the equation:
k * (q^2) / (0.3^2) * sin(θ) = 2 * (m * g / g)
k * (q^2) / (0.3^2) * sin(θ) = 2 * m
Finally, we substitute the value for the mass of the spheres:
k * (q^2) / (0.3^2) * sin(θ) = 2 * (2*(0.20 g) / g)
Now we can solve for the value of q.
To determine the amount of charge placed on each sphere, we can use Coulomb's Law, which states that the electrostatic force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's go through the steps to find the charge on each sphere:
1. Identify the given information:
- Mass of each metal sphere: 0.20 g
- Length of the non-conducting threads: 30 cm
- Angle between the threads: 20 degrees
2. Convert the mass of each sphere from grams to kilograms:
- Mass of each metal sphere: 0.20 g = 0.20 × 10^(-3) kg
3. Determine the gravitational force acting on each sphere:
- Gravitational force = mass × acceleration due to gravity
- Acceleration due to gravity ≈ 9.8 m/s^2
- Gravitational force = (0.20 × 10^(-3)) kg × 9.8 m/s^2
4. Determine the tension force in each thread:
- The system is in equilibrium, so the tension force in each thread balances the gravitational force acting on the respective sphere.
- Tension force = Gravitational force = (0.20 × 10^(-3)) kg × 9.8 m/s^2
5. Calculate the electrostatic force between the spheres:
- The electrostatic force acts as a repulsive force, pushing the spheres apart.
- To calculate the electrostatic force, we need the distance between the spheres.
- The distance between the spheres can be found using trigonometry, considering the length of the threads and the angle between them.
- Distance between the spheres = 2 × (length of thread) × sin(angle/2)
- Distance between the spheres = 2 × (0.30 m) × sin(20/2 degrees)
6. Substitute the values into Coulomb's Law equation:
- Electrostatic force = k × (charge on sphere1) × (charge on sphere2) / (distance between the spheres)^2
- The constant k is the electrostatic constant and its value is approximately 9 × 10^9 N·m^2/C^2.
7. Rearrange the equation to solve for the charge on each sphere:
- (charge on sphere1) × (charge on sphere2) = (Electrostatic force) × [(distance between the spheres)^2] / k
- Since both spheres have the same charge, we can consider (charge on sphere1) = (charge on sphere2) = q.
- q^2 = (Electrostatic force) × [(distance between the spheres)^2] / k
- q = sqrt[(Electrostatic force) × [(distance between the spheres)^2] / k]
8. Substitute the known values into the equation and calculate the charge on each sphere.
Please provide the value for the electrostatic force or any other relevant information if available to proceed with the calculations.