A charged particle is projected with velocityv0along positive x-axis. The magnetic fieldB is directed along negativez-axis betweenx= 0andx = L. The particle emerges out (atx = L) at an angle of 60° with the direction of projection. Find the velocity with which the same particle is projected (atx= 0)along positivex-axis so that when it emerges out (atx = L), the angle made by it is 30° with the direction of projection isnv0. Find the value ofn.Correct answer is '1.732'. Can you explain this answer?

Question

A charged particle is projected with velocityv0along positive x-axis. The magnetic fieldB is directed along negativez-axis betweenx= 0andx = L. The particle emerges out (atx = L) at an angle of 60&deg; with the direction of projection. Find the velocity with which the same particle is projected (atx= 0)along positivex-axis so that when it emerges out (atx = L), the angle made by it is 30&deg; with the direction of projection isnv0. Find the value ofn.Correct answer is '1.732'. Can you explain this answer?

Answer

To find the velocity of the charged particle at the point of emergence, we can use the principles of electromagnetic force.

The Lorentz force on a charged particle moving in a magnetic field is given by:

F = q(v x B)

where F is the force, q is the charge of the particle, v is the velocity vector of the particle, and B is the magnetic field vector.

In this case, the force is acting in the negative y-direction (since the particle emerges at an angle of 60 degrees), so we can write:

F = -q(v x B)

The magnitude of the force is given by:

|F| = q|v||B|sinθ

where θ is the angle between the velocity vector and the magnetic field vector.

Since the particle is projected along the positive x-axis, the velocity vector can be written as:

v = v0i

where v0 is the magnitude of the initial velocity and i is the unit vector along the x-axis.

The magnetic field vector is given to be directed along the negative z-axis, so we can write:

B = -Bk

where B is the magnitude of the magnetic field and k is the unit vector along the z-axis.

The angle θ between the velocity vector and the magnetic field vector is 90 degrees, since they are perpendicular to each other.

Plugging these values into the equation for the magnitude of the force, we get:

|F| = q|v0||B|sin90 = qv0B

The force is acting in the negative y-direction, so we can write:

F = -Fj

where j is the unit vector along the y-axis.

Since F = q(v x B), we can equate the corresponding components to get:

-F = q(v0B)j

Comparing the y-components, we get:

-qv0B = -F

Solving for v0, we find:

v0 = F/(qB)

So, the magnitude of the initial velocity is given by:

|v0| = |F|/(q|B|)

Now, we can substitute the given values:

|F| = mg (where m is the mass of the particle and g is the acceleration due to gravity)
q = charge of the particle
B = magnitude of the magnetic field

Plugging these values into the equation, we get:

|v0| = mg/(q|B|)

So, the magnitude of the initial velocity is mg/(q|B|).

Note: This calculation assumes that the only force acting on the particle is the electromagnetic force due to the magnetic field. Other forces such as gravitational force or electric force may also be present, depending on the specific scenario.

Similar Physics Doubts

A horizontal pipe carries water in the positive direction ofxincrease in the positive x-direction. Small identical holes are made at three different places along the pipe such that stream from them come out vertically upwards. The height upto which the water stream rise from the three holes

Top Courses for Physics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams