What is the magnitude of the minimum magnetic field that will keep the particle moving in the earth's gravitational field in the same horizontal, northward direction

The magnitude of the magnetic field vector is 1.91T and is directed towards the east.

The steps to the solution can be found in the attachment below.

Explanation:

For the charge to remain in the the earth' gravitational field the magnetic force on the charge must be equal to the earth's gravitational force on the charge and must act opposite the direction of the earth's gravitational force.

Fm = Fg

qvBSin(theta) = mg

Where q = magnitude of charge

v = magnitude of the velocity vector = 4 x10^4 m/s

B = magnitude of the magnetic field vector

theta = the angle between the magnetic field and velocity vectors = 90°

m = mass of the charge = 0.195g

g = acceleration due to gravity =9.8m/s²

On substituting the respective values of all variables in the equation (1) above

B = 1.91T

The direction of the magnetic field vector was found by the application of the right hand rule: if the thumb is pointed in the direction of the magnetic force and the index finger is pointed in the direction of the velocity vector, the middle finger points in the direction of the magnetic field.

Below is the step by step procedure to the solution.

the right answer is 0.3 meters per second