Consider three identical metal spheres, A, B and C. Sphere A carries of `-2.0 mu C`, spheres B carries a charge of `-6.0 mu C`, and spheres C carries a charge of `+5.0 mu C`. Sphere A and B are touched together and then separated. Sphere B and C are then touched and separated. Does sphere C end up with an excess or a deficiency of electrons and how many electrons is it ?

Options

A. deficiency, `6 xx 10^(13)`
B. excess, `3 xx 10^(13)`
C. excess, `2 xx 10^(13)`
D. deficiency, `3 xx 10^(12)`

Answer

To solve the problem step by step, we will analyze the charges on the spheres after they are touched together and then separated.

### Step 1: Initial Charges
- Sphere A: \( Q_A = -2.0 \, \mu C \)
- Sphere B: \( Q_B = -6.0 \, \mu C \)
- Sphere C: \( Q_C = +5.0 \, \mu C \)

### Step 2: Touching Sphere A and B
When two identical metal spheres are touched together, the total charge is shared equally between them.

1. **Calculate the total charge when A and B are touched:**
\[ Q_{total} = Q_A + Q_B = -2.0 \, \mu C + (-6.0 \, \mu C) = -8.0 \, \mu C \]

2. **Charge on each sphere after separation:**
Since they are identical, the charge on each sphere after touching will be:
\[ Q_{A'} = Q_{B'} = \frac{Q_{total}}{2} = \frac{-8.0 \, \mu C}{2} = -4.0 \, \mu C \]

### Step 3: Charges after touching A and B
- After touching and separating:
- Sphere A: \( Q_A' = -4.0 \, \mu C \)
- Sphere B: \( Q_B' = -4.0 \, \mu C \)
- Sphere C: \( Q_C = +5.0 \, \mu C \)

### Step 4: Touching Sphere B and C
Now, we touch Sphere B (which has \( -4.0 \, \mu C \)) and Sphere C (which has \( +5.0 \, \mu C \)).

1. **Calculate the total charge when B and C are touched:**
\[ Q_{total} = Q_B' + Q_C = -4.0 \, \mu C + 5.0 \, \mu C = +1.0 \, \mu C \]

2. **Charge on each sphere after separation:**
\[ Q_{B''} = Q_{C'} = \frac{Q_{total}}{2} = \frac{1.0 \, \mu C}{2} = +0.5 \, \mu C \]

### Step 5: Final Charges
- After touching and separating:
- Sphere A: \( Q_A' = -4.0 \, \mu C \)
- Sphere B: \( Q_B'' = +0.5 \, \mu C \)
- Sphere C: \( Q_C' = +0.5 \, \mu C \)

### Step 6: Determine Excess or Deficiency of Electrons in Sphere C
Initially, Sphere C had a charge of \( +5.0 \, \mu C \) and now it has \( +0.5 \, \mu C \).

1. **Calculate the change in charge:**
\[ \Delta Q_C = Q_C - Q_C' = 5.0 \, \mu C - 0.5 \, \mu C = 4.5 \, \mu C \]

This means Sphere C has lost \( 4.5 \, \mu C \) of positive charge, which corresponds to gaining \( 4.5 \, \mu C \) of negative charge (or electrons).

### Step 7: Calculate the Number of Electrons
To find the number of electrons gained:
1. **Charge of one electron:**
\[ e = 1.6 \times 10^{-19} \, C \]

2. **Calculate the number of electrons:**
\[ \text{Number of electrons} = \frac{4.5 \times 10^{-6} \, C}{1.6 \times 10^{-19} \, C/e^-} \approx 2.8125 \times 10^{13} \, e^- \]

Rounding this, we find that Sphere C ends up with approximately \( 2.8 \times 10^{13} \) excess electrons.

### Conclusion
Sphere C ends up with an excess of electrons, approximately \( 2.8 \times 10^{13} \) electrons.