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To solve this problem, we need to calculate the magnetic force that charge \( q_1 \) exerts on charge \( q_2 \). The force can be determined using the following equation from the Biot-Savart law:

\[
\mathbf{F} = q_2 \mathbf{v_2} \times \mathbf{B_1}
\]

Where:
- \( q_2 \) is the charge of the second particle,
- \( \mathbf{v_2} \) is the velocity vector of the second particle,
- \( \mathbf{B_1} \) is the magnetic field due to \( q_1 \) at the position of \( q_2 \).

The magnetic field \( \mathbf{B_1} \) due to charge \( q_1 \) is given by:

\[
\mathbf{B_1} = \frac{\mu_0}{4\pi} \frac{q_1 (\mathbf{v_1} \times \mathbf{r})}{r^3}
\]

Where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \) is the permeability of free space,
- \( q_1 \) is the charge of the first particle,
- \( \mathbf{v_1} \) is the velocity of the first particle,
- \( \mathbf{r} \) is the vector from \( q_1 \)'s position to \( q_2 \)'s position,
- \( r \) is the magnitude of \( \mathbf{r} \).

### Given values:
- \( q_1 = 4.20 \times 10^{-6} \, \text{C} \),
- \( q_2 = -3.10 \times 10^{-6} \, \text{C} \),
- \( \mathbf{v_1} = (9.20 \times 10^5 \, \hat{i}) \, \text{m/s} \),
- \( \mathbf{v_2} = (-5.30 \times 10^5 \, \hat{j}) \, \text{m/s} \),
- \( \mathbf{r} = (0.150 \, \hat{i} - 0.250 \, \hat{j}) \, \text{m} \),
- \( r = \sqrt{(0.150)^2 + (0.250)^2} = 0.2915 \, \text{m} \).

Let's proceed by calculating the necessary vectors and values to find the magnetic force \( \mathbf{F} \).The magnetic force that \( q_1 \) exerts on \( q_2 \) is approximately:

\[
\mathbf{F} = (-6.40 \, \mu\text{N}, 0 \, \mu\text{N}, 0 \, \mu\text{N})
\]

Thus, the force is primarily in the \( x \)-direction with a magnitude of \( -6.40 \, \mu\text{N} \).

Would you like further explanation or details on any step? Here are some related questions for further practice:

1. How would the force change if the velocity of \( q_1 \) were doubled?
2. What would happen to the magnetic force if both charges had the same sign?
3. How does the magnetic force depend on the distance between the charges?
4. Can you explain why the force has no component in the \( y \)- or \( z \)-directions?
5. How would the result differ if the velocity of \( q_2 \) was reversed?

**Tip:** When dealing with cross products, remember that the result is always perpendicular to both original vectors.

At this instant, what is the magnetic force that q1 exerts on q2?

Find the magnetic force that charge q1 exerts on charge q2 using vector calculus and the Biot-Savart law. Learn how to apply the formulas for electromagnetism with given charges, velocities, and distances. Suitable for university-level physics students.

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