Final answer:
The problem is solved by understanding that the magnetic force acts as a centripetal force causing circular motion, and equating the two forces allows for the determination of the specific charge of the particle, which turns out to be option B: Bd/vcosθ.
Step-by-step explanation:
The question involves finding the specific charge (charge-to-mass ratio, q/m) of a charged particle that is deflected by a uniform magnetic field when moving along the positive x-direction. When the particle with velocity v enters the region with a magnetic field →B, which is directed in the negative k-direction (out of the page), it experiences a magnetic force perpendicular to both its velocity and the magnetic field direction. This force causes the particle to move in a circular path in the plane perpendicular to the field, and the particle gets deflected at an angle θ from its initial path.
Since the force acts perpendicular to the velocity of the particle, the magnetic force supplies the centripetal force necessary for circular motion which is given by the equation Fc = mv2/r, while the magnetic force is given by F = qvB, where q is the charge of the particle, m is its mass, B is the magnetic field strength, and r is the radius of the circular path.
By equating the magnetic force to the centripetal force (F = Fc), we can find the specific charge of the particle as q/m = v/rB. The radius r can be determined from the geometry of the particle's deflection and the definition of the angle θ, which is related to the particle's path. In this scenario, the particle's path exits tangentially to the circle it would make if it were in the field indefinitely. Thus, due to the deflection, the radius can be geometrically related to the distance d the particle has traveled in the field and the angle of deflection θ.
Combining these relations and solving for the specific charge will give us the correct answer, which is option B: Bd/vcosθ. The use of trigonometric relationships and understanding of circular motion in a magnetic field are crucial to solving this problem.