Metal sphere 1 has a positive charge of 7.00nC . Metal sphere 2, which is twice the diameter of sphere 1, is initially uncharged. The spheres are then connected together by a long, thin metal wire. What are the final charges on each sphere?
A.) Q1' = (Q total/r total)*r1 = (7/3r)*r = 7/3 = 2.33 nC
B.) Q2' = (Q total/r total)*r1 = (7/3r)*2r = 14/3 = 4.67 nC
A.) Q1' = (Q total/r total)*r1 = (5/3r)*r = 5/3 = 2.33 nC
B.) Q2' = (Q total/r total)*r1 = (5/3r)*2r = 10/3 = 4.67 nC
To find the final charges on each sphere, we can use the Law of Conservation of Charge, which states that the total charge in a closed system remains constant.
Let's denote the charge on sphere 1 as Q1 and the charge on sphere 2 as Q2.
Step 1: Find the charge on sphere 2 when it is connected to sphere 1.
Since sphere 2 is initially uncharged, its initial charge, Q2, is zero.
Step 2: Calculate the total charge after the spheres are connected.
According to the Law of Conservation of Charge, the total charge remains constant when the spheres are connected. Therefore, we have:
Q1 + Q2 = Total charge
Step 3: Substitute the known values into the equation.
Q1 + 0 = Total charge
Q1 = Total charge
Since the spheres are connected, they equilibrate charge. This means that their charges will be equal after connection.
Step 4: Determine the total charge on the connected spheres.
From the given information, sphere 1 has a positive charge of 7.00 nC.
Therefore, the total charge is equal to the charge on sphere 1, which is 7.00 nC.
Step 5: Divide the total charge equally between the two spheres.
Since the spheres equilibrate charge, they will share the total charge equally. Thus:
Q1 = Total charge / 2
Q1 = 7.00 nC / 2
Q1 = 3.50 nC
Therefore, the final charges on each sphere are:
Sphere 1: 3.50 nC
Sphere 2: 3.50 nC
To determine the final charges on each sphere, we need to understand the principle of charge conservation. According to this principle, when two conductive objects are connected (in this case, metal spheres), charge redistributes until both objects reach the same potential.
Here's how you can calculate the final charges:
1. Start by determining the initial charge on each sphere:
- Sphere 1 has a positive charge of 7.00 nC.
- Sphere 2 is initially uncharged, so it has a charge of 0 nC.
2. Since the spheres are connected by a wire, charge will flow from the positively charged sphere to the initially uncharged sphere until equilibrium is reached.
3. Consider the fact that both spheres are conductive, and when conductive objects touch, they exchange charge until they reach the same potential.
4. Since sphere 2 has twice the diameter of sphere 1, it has four times the surface area. Therefore, it can hold four times the charge that sphere 1 can hold.
5. Calculate the amount of charge transferred from sphere 1 to sphere 2:
- Divide the initial charge of sphere 1 (7.00 nC) by 5 (since 4 parts go to sphere 2 and 1 part remains with sphere 1). This gives you (7.00 nC) / 5 = 1.40 nC.
- Note that we divide by 5 because 4 parts go to sphere 2 (sphere 2 can hold four times the charge) and 1 part remains with sphere 1.
6. Calculate the final charge on each sphere:
- Subtract the charge transferred from sphere 1 from its initial charge: 7.00 nC - 1.40 nC = 5.60 nC.
- Add the charge transferred from sphere 1 to sphere 2: 0 nC + 1.40 nC = 1.40 nC.
Therefore, the final charges on the spheres are:
- Sphere 1: 5.60 nC (positive charge)
- Sphere 2: 1.40 nC (positive charge)