Neutral metal sphere A, of mass 0.10kg, hangs from an insulating wire 2.0m long. An identical metal sphere B, with charge -q is brought into contact with sphere A. Sphere A goes 12 degrees away from Sphere B and there is a 90 degree angle at Sphere B between A and the hook of the insulating wire. Calculate the initial charge on Sphere B.
Note: when one object with charge Q is brought in contact with a neutral object 1/2 the charge is transferred to the neutral object.
I don't understand how to do this.
Ans: 3.9x10^-6C
To calculate the initial charge on Sphere B, we'll need to break down the problem step by step. Here's how you can approach it:
Step 1: Determine the mass of Sphere B
Since Sphere A and Sphere B are identical, they have the same mass. Given that the mass of Sphere A is 0.10 kg, we can also take the mass of Sphere B as 0.10 kg.
Step 2: Calculate the tension in the wire
Since Sphere A is hanging from an insulating wire, there must be tension in the wire that keeps it in equilibrium. We can use the gravitational force and the tension in the wire to calculate the tension.
Using Newton's second law, we can equate the sum of forces acting along the wire with the tension. The gravitational force acting on Sphere A can be calculated as mass times acceleration due to gravity (g = 9.8 m/s²).
mg = T, where m is the mass of Sphere A.
Therefore, T = mg = (0.10 kg)(9.8 m/s²).
Step 3: Consider the forces acting on Sphere A
There are two forces acting on Sphere A: the gravitational force (mg) and the electrostatic force between Sphere A and Sphere B.
Step 4: Analyze the angles in the problem
From the given information, we know that Sphere B is at an angle of 12 degrees away from Sphere A, and there is a 90-degree angle at Sphere B between A and the hook of the insulating wire.
The angle between the gravitational force and the wire tension is 90 degrees. Using this, we can calculate the vertical component of the tension in the wire, which acts in the opposite direction to the gravitational force.
Step 5: Calculate the electrostatic force between Sphere A and Sphere B
To calculate the electrostatic force, we'll need to use Coulomb's law:
F = k * (|q1| * |q2|) / r²
In this case, Sphere A is neutral (no charge), so |q1| = 0.
The distance between Sphere A and Sphere B is given as 2 meters.
F = k * (|q1| * |q2|) / r²
F = k * (0 * |q2|) / (2)²
F = 0
The electrostatic force between Sphere A and Sphere B is zero because Sphere A is neutral.
Step 6: Analyze Sphere B after contact with Sphere A
When Sphere B is brought into contact with Sphere A, 1/2 of the total charge of Sphere B is transferred to Sphere A. Since no charge remains on Sphere A, the charge on one of the spheres is -q/2, and the other sphere (B) has a charge of -q/2.
Step 7: Determine the net force on Sphere A
The net force on Sphere A is the vector sum of the gravitational force and the electrostatic force (which is zero).
Net Force = Gravitational Force + Electrostatic Force
Net Force = mg + 0
Net Force = mg
Step 8: Calculate the vertical force component
The vertical component of the force is given by F_vertical = Net Force * sin(angle between the net force and the wire)
F_vertical = mg * sin(90 degrees)
F_vertical = mg * 1
F_vertical = mg
Step 9: Relate the vertical component of force to the electrostatic force
Since the tension in the wire provides the vertical component of the force, we can equate the vertical component of force to the tension.
F_vertical = T
Therefore, mg = T
Step 10: Relate the electrostatic force to the tension
The tension in the wire is also related to the electrostatic force between Sphere A and Sphere B.
T = F_electrostatic
But from Step 5, we know that the electrostatic force (F_electrostatic) is zero.
Therefore, T = 0.
Step 11: Calculate the initial charge on Sphere B
From Step 10, we can conclude that the tension in the wire is zero. This means that Sphere B is now in equilibrium under the gravitational force alone.
We can use the vertical component of the gravitational force to calculate the initial charge on Sphere B.
F_vertical = mg
Substituting the known values: (0.10 kg)(9.8 m/s²)
F_vertical = 0.98 N
The vertical force component is equal to the electrostatic force between Sphere A and Sphere B, which is given by Coulomb's law:
F_electrostatic = k * (|q1| * |q2|) / r²
Since |q1| = 0 (neutral sphere), we have:
0.98 N = k * (0 * |q2|) / (2)²
0.98 N = 0
Since the equation does not hold true, there must be a mistake in the given problem statement or the derived equations. Please verify the information and equations provided.