Question

Figure 28-25 shows the path of a particle through six regions of uniform magnetic field, where the path is either a half-circle or a quarter-circle. Upon leaving the last region, the particle travels between two charged, parallel plates and is deflected toward the plate of higher potential. What is the direction of the magnetic field in each of the six regions?

Short answer

In region a, d, and e, the magnetic field goes into the plane and in region b, c, and f, it is out of the plane.

Step-by-step solution

Given

The magnetic field is uniform in all six regions.

Determining the concept

Use the equation of magnetic force to find the direction of the magnetic field.

Right Hand Rule states that if we arrange our thumb, forefinger and middle finger of the right-hand perpendicular to each other, then the thumb points towards the direction of the motion of the conductor relative to the magnetic field, the forefinger points towards the direction of the magnetic field and the middle finger points towards the direction of the induced current.

Formulae are as follow:

$F=q\left(\overrightarrow{v}\times \overrightarrow{B}\right)=qvB\sin \theta$

Where, F is magnetic force, v is velocity, B is magnetic field, q is charge of particle.

Determining the direction of magnetic field in each of the six regions

From the diagram, the particle is deflected towards the plate with a higher potential. Hence the particle must be a negatively charged particle.

In region (a):

The particle travels a half circle, and the force acting on the particle is in the right direction, and the velocity of the particle is upward. Using the right-hand rule, the magnetic field is in the inward direction to the plane of the paper.

In region (b)

The force acting on the particle is towards the right, and the velocity is downward. Using the right-hand rule, the direction ofthemagnetic field is outward.

In region (c)

The direction of the magnetic field is outward because the centripetal force acting on the particle is towards the left, and velocity is upward.

In region (d)

The force is upwards, and the velocity is towards left. Using the right-hand rule, the magnetic field is in the inward direction.

In region (e)

The direction of the magnetic field is inward because the force acting on the particle is downward, and velocity is towards the right.

In region (f)

The velocity direction is downward. The force acts to the right. So, the direction of the magnetic field is out of the page.

Hence, in region a, d, and e, the magnetic field goes into the plane, and in regions b, c, and f, it is out of the plane.