Two metallic spheres of mass .001 kg hang from a point by .3 m strings. Each is given the same amount and sign of charge. After equilibrium is reached, they each make an angle of 20 degrees relative to the vertical. Determine the amount of charge in each sphere.
Before we start the distance between the balls is .205m
Tension in string
Tcos20 = mg
so T = mg/cos20
Solve for T
Tsin20 = kq^2/r which we found from geometry.
Solve for q
A picture really helps with these...
To determine the amount of charge in each sphere, we can use the principle of electrostatic equilibrium. In this case, we have two charged spheres in equilibrium under the influence of gravity and the electric force.
Let's use some variables to represent the unknowns:
- q1: Charge of the first sphere
- q2: Charge of the second sphere
We know the following information:
- Mass of each sphere: 0.001 kg
- Length of the strings: 0.3 m
- Angle made by each sphere with the vertical: 20 degrees
When the spheres are in equilibrium, the electrical force acting on each sphere must be equal and opposite to the weight of the sphere. We can write the following equation for each sphere:
Electrical force = Weight
Now, let's break down these forces:
1. Weight:
The weight of each sphere can be calculated using the equation:
Weight = mass × acceleration due to gravity
Given that the mass of each sphere is 0.001 kg and acceleration due to gravity is 9.8 m/s^2, we can calculate the weight of each sphere.
2. Electrical force:
The electrical force between two charged spheres is given by Coulomb's Law:
Electrical force = (k × |q1 × q2|) / distance^2
In this case, the distance between the spheres is twice the length of the string because they hang at an angle of 20 degrees.
3. Angle and distance
We are given that each sphere makes a 20-degree angle with the vertical. The vertical component of the distance can be calculated as:
Vertical distance = length of the string × sin(angle)
4. Equilibrium condition
Since the spheres are in equilibrium, the electrical force acting on each sphere should be equal and opposite. So we can equate the magnitudes of the electrical forces in our equation.
Now, let's put it all together:
1. Calculate the weight of each sphere:
Weight = mass × acceleration due to gravity
2. Calculate the vertical distance for each sphere:
Vertical distance = length of the string × sin(angle)
3. Calculate the distance between the spheres:
Distance = 2 × vertical distance
4. Use Coulomb's Law to equate the electrical forces:
(k × |q1 × q2|) / distance^2 = Weight
5. Solve the equation for q1 and q2.