Question

A particle with a charge of − 6.00 nC is moving in a uniform magnetic field of B⃗ =−( 1.21 T )k^ . The magnetic force on the particle is measured to be F⃗ =−( 3.40×10−7 N )i^+( 7.60×10−7 N )j^ . Calculate the x component of the velocity of the particle.

Answer 1

The x component of the velocity of the particle is -5.71 m/s. The formula for the magnetic force on a moving charged particle in a magnetic field is used to calculate the velocity component. The given x component of the force and the charge of the particle are used to solve for the velocity.

Step-by-step explanation:

The x component of the velocity of the particle can be found by using the formula for the magnetic force on a moving charged particle in a magnetic field. The formula is F = qvBsinθ, where F is the force, q is the charge of the particle, v is the velocity of the particle, B is the magnetic field, and θ is the angle between the velocity and the magnetic field. Since the force is measured to be F⃗ =−( 3.40×10−7 N )i^+( 7.60×10−7 N )j^, we can separate the force into its x and y components: Fx = -3.40×10−7 N and Fy = 7.60×10−7 N.

We can then set up two equations using the x and y components of the force and rearrange them to solve for the x component of the velocity, vx. From the x component equation, we have Fx = qvxB, which can be rearranged as qx = Fx / (vB). Substituting the given values, we have -6.00×10^(-9) C = -3.40×10^(-7) N / (vB). From the y component equation, we have Fy = qvyB, which can be rearranged as qy = Fy / (vB). Substituting the given values, we have 0 = 7.60×10^(-7) N / (vB). Solving both equations simultaneously, we find that vx = -5.71 m/s.

Answer 2

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