Let's solve the given problem step by step:
### Constants:
- Coulomb's constant (k): [tex]\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)[/tex]
- Charge of a proton (q): [tex]\( 1.6 \times 10^{-19} \, \text{C} \)[/tex]
### Distances:
- Distance to point A (r_A): [tex]\( 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)[/tex]
- Distance to point B (r_B): [tex]\( 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 1 \times 10^{-4} \, \text{m} \)[/tex]
### (a) Electric potential at points A and B:
The electric potential [tex]\( V \)[/tex] at a point due to a point charge is given by:
[tex]\[ V = \frac{kq}{r} \][/tex]
1. Electric potential at point A ([tex]\( V_A \)[/tex]):
[tex]\[ V_A = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \times 1.6 \times 10^{-19} \, \text{C}}{1 \times 10^{-3} \, \text{m}} \][/tex]
Calculating [tex]\( V_A \)[/tex]:
[tex]\[ V_A \approx 1.4384 \times 10^{-6} \, \text{V} \][/tex]
2. Electric potential at point B ([tex]\( V_B \)[/tex]):
[tex]\[ V_B = \frac{8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \times 1.6 \times 10^{-19} \, \text{C}}{1 \times 10^{-4} \, \text{m}} \][/tex]
Calculating [tex]\( V_B \)[/tex]:
[tex]\[ V_B \approx 1.4384 \times 10^{-5} \, \text{V} \][/tex]
### (b) Electric potential difference between point A and point B:
The electric potential difference [tex]\( \Delta V \)[/tex] between two points is given by:
[tex]\[ \Delta V = V_B - V_A \][/tex]
Calculating [tex]\( \Delta V \)[/tex]:
[tex]\[ \Delta V = 1.4384 \times 10^{-5} \, \text{V} - 1.4384 \times 10^{-6} \, \text{V} \][/tex]
[tex]\[ \Delta V \approx 1.2946 \times 10^{-5} \, \text{V} \][/tex]
### Summary of Results:
1. Electric potential at point A ([tex]\( V_A \)[/tex]): [tex]\( 1.4384 \times 10^{-6} \, \text{V} \)[/tex]
2. Electric potential at point B ([tex]\( V_B \)[/tex]): [tex]\( 1.4384 \times 10^{-5} \, \text{V} \)[/tex]
3. Electric potential difference ([tex]\( \Delta V \)[/tex]): [tex]\( 1.2946 \times 10^{-5} \, \text{V} \)[/tex]
These results show the electric potentials and the potential difference at the specified points due to a proton.
