. Four identical metal spheres have charges of qA = –8.0 µC, qB =–2.0 µC, qC = +5.0µ C, and qD = +12.0µ C.
(a) Two of the spheres are brought together so they touch, and then they are separated. Which spheres are they, if the final charge on each one is +5.0 µC?
(b) In a similar manner, which three spheres are brought together and then separated, if the final charge on each of the three is +3.0 µC?
(a) qB =–2.0 µC and qD = +12.0µ C.
q=(- 2 + 12)/2 = 5 µC
(b) qA = –8.0 µC, qC = +5.0µ C, and qD = +12.0µ C.
q=(- 8 + 5+12)/3 = 9/3=3 µC
(a) To determine which spheres are brought together, we need to consider the conservation of charge.
When two spheres were brought together and then separated, the charge on each sphere after the separation is +5.0 µC.
Let's assume that the two spheres brought together initially were sphere A and sphere B.
When they touch and separate, the total charge is conserved, meaning that the sum of the charges on the two spheres after separation should be equal to the sum of the charges on the two spheres before they touched.
qA + qB = qA' + qB'
Substituting the given values:
-8.0 µC + (-2.0 µC) = 5.0 µC + 5.0 µC
-10.0 µC = 10.0 µC
Since the equation is not satisfied, we know that spheres A and B are not brought together.
Next, let's assume that spheres A and C are brought together initially.
qA + qC = qA' + qC'
-8.0 µC + 5.0 µC = 5.0 µC + 5.0 µC
-3.0 µC = 10.0 µC
Again, the equation is not satisfied, so spheres A and C are not brought together.
Finally, let's assume that spheres A and D are brought together initially.
qA + qD = qA' + qD'
-8.0 µC + 12.0 µC = 5.0 µC + 5.0 µC
4.0 µC = 10.0 µC
Since the equation is satisfied, we conclude that spheres A and D are brought together initially.
Therefore, the two spheres brought together and then separated are sphere A and sphere D.
(b) Following a similar approach, let's assume that the three spheres brought together initially are sphere A, sphere B, and sphere C.
qA + qB + qC = qA' + qB' + qC'
-8.0 µC + (-2.0 µC) + 5.0 µC = 3.0 µC + 3.0 µC + 3.0 µC
-5.0 µC = 9.0 µC
The equation is not satisfied, so spheres A, B, and C are not brought together initially.
Therefore, the three spheres brought together and then separated are not determined, as there is no combination that satisfies the given conditions.
To solve this problem, we need to apply the principle of charge conservation, which states that the total charge before and after any interaction remains the same.
(a) Let's solve part (a) first.
Step 1: Determine the total charge before any interaction.
The total charge before any interaction is given by qA + qB + qC + qD.
Total charge before interaction = qA + qB + qC + qD
= -8.0 µC + (-2.0 µC) + 5.0 µC + 12.0 µC
= 7.0 µC
Step 2: Determine the total charge after the spheres are brought together and separated.
When two spheres touch, they equalize their charges, so they end up with the same charge. Therefore, two of the spheres must have a final charge of +5.0 µC.
Let's assume qA and qB are the two spheres that are brought together.
Now, we know that qA = qB = +5.0 µC.
Step 3: Determine the total charge after the spheres are separated.
Since the final charge on each one is +5.0 µC, the total charge after separation should still be equal to the total charge before interaction.
Let's assume qC and qD are the spheres that are brought together and then separated.
So, qA + qB + qC + qD = 7.0 µC
qA + qB + qC + qD = +5.0 µC + +5.0 µC + qC + qD = 7.0 µC
Simplifying the equation, we find:
10.0 µC + qC + qD = 7.0 µC
qC + qD = -3.0 µC
Now, we have two equations:
qA = qB = +5.0 µC
qC + qD = -3.0 µC
Comparing these two equations, we can conclude that the spheres which are brought together and separated are qC and qD.
Therefore, the answer to part (a) is: qC and qD are the spheres that are brought together and then separated.
(b) Now let's solve part (b).
Step 1: Determine the total charge before any interaction.
The total charge before any interaction is the same as in part (a) and given by qA + qB + qC + qD.
Total charge before interaction = qA + qB + qC + qD
= -8.0 µC + (-2.0 µC) + 5.0 µC + 12.0 µC
= 7.0 µC
Step 2: Determine the total charge after the spheres are brought together and separated.
When three spheres touch, they equalize their charges, so they end up with the same charge. Therefore, three of the spheres must have a final charge of +3.0 µC.
Let's assume qA, qB, and qC are the three spheres that are brought together.
Now, we know that qA = qB = qC = +3.0 µC.
Step 3: Determine the total charge after the spheres are separated.
Since the final charge on each one is +3.0 µC, the total charge after separation should still be equal to the total charge before interaction.
Let's assume qD is the sphere that is brought together and then separated.
So, qA + qB + qC + qD = 7.0 µC
qA + qB + qC + qD = +3.0 µC + +3.0 µC + +3.0 µC + qD = 7.0 µC
Simplifying the equation, we find:
9.0 µC + qD = 7.0 µC
qD = -2.0 µC
Now, we have two equations:
qA = qB = qC = +3.0 µC
qD = -2.0 µC
Comparing these two equations, we can conclude that the spheres which are brought together and separated are qD.
Therefore, the answer to part (b) is: qD is the sphere that is brought together and then separated.