A particle having mass m and charge q is released from the origin in a region in which electric field and magnetic field are given by B= -a j, E= b k, Find the y component of the velocity and the speed of the particle as a function of its z coordinate?




In this problem, we are given the electric and magnetic fields in a region where a charged particle is released. We need to find the y-component of the velocity and the speed of the particle as a function of its z-coordinate.


The electric and magnetic fields are given by:

- B = -a j
- E = b k

Where a and b are constants.


When a charged particle is released in this region, it experiences both electric and magnetic forces. The force on the charged particle is given by the Lorentz force equation:

- F = q(E + v x B)

Where v is the velocity of the charged particle.

The y-component of the velocity can be found by using the equation of motion:

- v_y = v_0 + a_y t

Where v_0 is the initial velocity, a_y is the acceleration in the y-direction, and t is the time.


The force on the charged particle in the y-direction is zero as there is no component of the magnetic field in the y-direction. Therefore, the acceleration in the y-direction is zero and the y-component of the velocity remains constant.

The force on the charged particle in the z-direction is given by:

- F_z = q(E_z)

= q(b)

Therefore, the acceleration in the z-direction is given by:

- a_z = F_z/m

= qb/m

The velocity in the z-direction can be found by using the equation of motion:

- v_z = v_{z0} + a_z t

Where v_{z0} is the initial velocity in the z-direction.

The speed of the charged particle is given by:

- v = sqrt(v_y^2 + v_z^2)

= sqrt((constant)^2 + (v_{z0} + qb/m t)^2)


The y-component of the velocity remains constant and the z-component of the velocity increases linearly with time. The speed of the charged particle increases with time and is a function of the z-coordinate.

Since magnetic field doesn't affect the velocity.so, it will be affected only by electric field.
force due to electric field=mA=qE=qb

therefore, acceleration of the partical =A =qb/m and this is constant.
also initial velocity of the partical is zero

therefore, V=√2Az. where, z denotes displacement along z-axis bcoz electric field is in that direction.

therefore, v=√[2(qb/m)z