Final answer:
To find the x component of the velocity of a charged particle in a magnetic field, use the second relationship Fy = -qvxBz and the provided values of force, charge, and magnetic field strength. By substituting these values and solving for vx, we find that the x component of the particle's velocity equals 105.37 m/s.
Step-by-step explanation:
To calculate the x component of the velocity (vx) of a charged particle moving in a magnetic field when given the magnetic force, we can use the formula for the magnetic force F on a moving charge, which can be expressed as F = qv x B, where q is the charge, v is the velocity of the particle, and B is the magnetic field. The magnetic force is measured as a vector F = (Fx, Fy, Fz), and the velocity of the particle is also a vector v = (vx, vy, vz).
In this scenario, the magnetic field is given as B = -(1.21 T)k^, which means it is directed in the negative z-direction. The magnetic force vector is given as F = -(3.40 x 10-7 N)i^ + (7.60 x 10-7 N)j^. Since there is no z-component (no Fz) for the force, this implies that there is no velocity in the z-direction (vz = 0) because the force from a magnetic field is always perpendicular to the direction of the particle's velocity.
From the relationship between force and velocity in a magnetic field, we can determine that Fx = qvyBz and Fy = -qvxBz. Given that Bz = -1.21 T, q = -6.00 nC (which is -6.00 x 10-9 C), and Fy = 7.60 x 10-7 N, we can isolate and solve for vx to find the x component of the velocity. We use Fy because it is influenced by vx due to the right-hand rule for cross products.
Using the second relationship and substituting the known values, we get:
Fy = -qvxBz
7.60 x 10-7 N = -(-6.00 x 10-9 C)vx(-1.21 T)
Solving for vx gives:
vx = (7.60 x 10-7 N) / (6.00 x 10-9 C x 1.21 T) = 105.37 m/s