Electromagnetism

Step 1: Lorentz Force Law
The force on a charged particle moving with velocity in a magnetic field is given by the Lorentz force law:

where is the charge of the particle. Since the force is given as , the components of the force must come from the cross product of and .
Step 2: Compute the Cross Product

Expanding the determinant, we get:

Step 3: Compare Components
Comparing the components of with the expression for , we get:

These equations imply relationships between the components of and , specifically:

Step 4: Conclusion
From this, we conclude that the magnetic field must have components in all three directions , and the correct expression for is: