A point charge `q_1=+2.40muC` is held stationary at the origin. A second point charge `q_2=-4.30muC` moves from the point x=0.150 m, y=0 to the point x=0.250 m, y=0.250 m. How much work is done by the electric force on `q_2`?

Explanation

To solve the problem of calculating the work done by the electric force on the charge \( q_2 \) as it moves from its initial position to its final position, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges and Their Positions**:
- The stationary charge \( q_1 = +2.40 \, \mu C = 2.40 \times 10^{-6} \, C \) is located at the origin (0,0).
- The moving charge \( q_2 = -4.30 \, \mu C = -4.30 \times 10^{-6} \, C \) starts at the position \( (0.150 \, m, 0) \) and moves to the position \( (0.250 \, m, 0.250 \, m) \).

2. **Calculate the Initial and Final Distances**:
- The initial distance \( r_{initial} \) from \( q_1 \) to \( q_2 \) when \( q_2 \) is at \( (0.150, 0) \):
\[ r_{initial} = \sqrt{(0.150 - 0)^2 + (0 - 0)^2} = 0.150 \, m \]
- The final distance \( r_{final} \) from \( q_1 \) to \( q_2 \) when \( q_2 \) is at \( (0.250, 0.250) \):
\[ r_{final} = \sqrt{(0.250 - 0)^2 + (0.250 - 0)^2} = \sqrt{0.250^2 + 0.250^2} = 0.250 \sqrt{2} \approx 0.3536 \, m \]

3. **Calculate the Initial and Final Potential Energies**:
- The formula for the potential energy \( U \) between two point charges is given by:
\[ U = k \frac{q_1 q_2}{r} \]
- Where \( k = 8.99 \times 10^9 \, N \cdot m^2/C^2 \).
- Calculate the initial potential energy \( U_{initial} \):
\[ U_{initial} = k \frac{q_1 q_2}{r_{initial}} = (8.99 \times 10^9) \frac{(2.40 \times 10^{-6})(-4.30 \times 10^{-6})}{0.150} \]
- Calculate the final potential energy \( U_{final} \):
\[ U_{final} = k \frac{q_1 q_2}{r_{final}} = (8.99 \times 10^9) \frac{(2.40 \times 10^{-6})(-4.30 \times 10^{-6})}{0.250 \sqrt{2}} \]

4. **Calculate the Work Done by the Electric Force**:
- The work done \( W \) by the electric force is equal to the negative change in potential energy:
\[ W = U_{initial} - U_{final} \]

5. **Substituting Values**:
- Substitute the values calculated in the previous steps to find \( W \).