Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?

To find the number of electrons in the speck of dust, let's break down the problem:

1. **Total charge from protons**:
   - Each proton has a charge of \(+1.602 \times 10^{-19}\) coulombs.
   - The speck of dust has \(1.0000 \times 10^{12}\) protons.
   - Total positive charge from the protons is:
     \[
     Q_{\text{protons}} = (1.0000 \times 10^{12}) \times (1.602 \times 10^{-19} \text{ C}) = 1.602 \times 10^{-7} \text{ C}
     \]

2. **Net charge of the speck**:
   - The net charge of the speck is given as \(-5.00\) nC.
   - Convert this to coulombs:
     \[
     Q_{\text{net}} = -5.00 \times 10^{-9} \text{ C}
     \]

3. **Calculate the charge due to the electrons**:
   - Let the charge due to electrons be \(Q_{\text{electrons}}\).
   - The total charge on the speck is the sum of the charges from protons and electrons:
     \[
     Q_{\text{net}} = Q_{\text{protons}} + Q_{\text{electrons}}
     \]
   - Since the net charge is negative, we expect \(Q_{\text{electrons}}\) to be negative and larger in magnitude than \(Q_{\text{protons}}\).
     \[
     -5.00 \times 10^{-9} \text{ C} = 1.602 \times 10^{-7} \text{ C} + Q_{\text{electrons}}
     \]
   - Solve for \(Q_{\text{electrons}}\):
     \[
     Q_{\text{electrons}} = -5.00 \times 10^{-9} \text{ C} - 1.602 \times 10^{-7} \text{ C} = -1.652 \times 10^{-7} \text{ C}
     \]

4. **Calculate the number of electrons**:
   - The charge of one electron is \(-1.602 \times 10^{-19}\) C.
   - The number of electrons \(n\) can be found by:
     \[
     n = \frac{Q_{\text{electrons}}}{\text{Charge per electron}} = \frac{-1.652 \times 10^{-7} \text{ C}}{-1.602 \times 10^{-19} \text{ C/electron}}
     \]
     \[
     n \approx 1.031 \times 10^{12} \text{ electrons}
     \]

So, the speck of dust has approximately \(1.031 \times 10^{12}\) electrons.

Would you like more details or have any questions? Here are 5 related questions:

1. How would the number of electrons change if the net charge were \(-10.00\) nC?
2. What would be the net charge if the speck had \(1.2000 \times 10^{12}\) protons and the same number of electrons as calculated above?
3. How would you calculate the number of protons in a speck if only the net charge and number of electrons were known?
4. If a similar speck had a net charge of \(+5.00\) nC, how many electrons would it have compared to the current situation?
5. What physical factors might cause a dust speck to gain or lose electrons?

**Tip**: Always double-check the signs and units when working with electrostatic charges to avoid errors in the final result.

Learn how to calculate the number of electrons in a dust speck with a net charge of -5.00 nC. Understand the relationship between proton count, net charge, and electron count using fundamental principles of electric charge. Suitable for advanced high school students and beyond.

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