In a certain region uniform electric flied, E= -E'k and magnetic field, B=B'k are present. at time t=0, a particle of mass m and charge q is given a velocity v= v'j+v'k. Find the min. speed of the particle and the time when happens so.

pls temme how to proceed...and also duz B=B'k mean the particle travels on the z-axis itself?

To tackle this problem, we need to analyze the motion of a charged particle in the presence of both electric and magnetic fields. The given fields are uniform, with the electric field represented as **E = -E'k** and the magnetic field as **B = B'k**. The particle starts with an initial velocity of **v = v'j + v'k**. Let's break this down step by step.

Understanding the Forces Acting on the Particle

When a charged particle moves through electric and magnetic fields, it experiences forces due to both fields. The electric force (**F_e**) acting on the particle is given by:

• F_e = qE

Substituting the expression for the electric field:

• F_e = q(-E'k) = -qE'k

Next, the magnetic force (**F_m**) is given by the Lorentz force law:

• F_m = q(v × B)

Here, the velocity vector **v = v'j + v'k** and the magnetic field **B = B'k**. The cross product can be calculated as follows:

• v × B = (v'j + v'k) × (B'k)

Using the right-hand rule and the properties of cross products, we find:

• v × B = v'B'(j × k) = -v'B'i

Thus, the magnetic force becomes:

• F_m = q(-v'B'i) = -qv'B'i

Net Force and Motion Analysis

The total force acting on the particle is the sum of the electric and magnetic forces:

• F_total = F_e + F_m = -qE'k - qv'B'i

This indicates that the particle experiences forces in both the negative z-direction and the negative x-direction. The motion can be analyzed in two dimensions: the x-y plane (where the particle moves due to the initial velocity) and the z-direction (where the electric field acts).

Finding the Minimum Speed

To find the minimum speed of the particle, we need to consider the balance of forces. The particle will experience a drift due to the electric field, which will affect its trajectory. The motion in the z-direction will be influenced by the electric field, while the x-direction will be influenced by the magnetic field.

At equilibrium, the forces in the x-direction must balance out. This occurs when:

• qE' = qv'B'

From this, we can derive the condition for the minimum speed:

• E' = v'B'

Thus, the minimum speed **v'** can be expressed as:

• v' = E' / B'

Determining the Time of Minimum Speed

To find the time when this minimum speed occurs, we can use the relationship between velocity, acceleration, and time. The acceleration in the z-direction due to the electric field is:

• a_z = F_e/m = -qE'/m

Assuming the particle starts from rest in the z-direction, we can use the kinematic equation:

• z = 0.5 * a_z * t^2

Setting the z-position equal to the distance traveled under the influence of the electric field will allow us to solve for time. However, since the particle is also moving in the x-direction due to the magnetic field, we need to consider the trajectory in both dimensions to find the exact time when the minimum speed occurs.

Clarifying the Motion Along the Axes

Regarding your question about whether **B = B'k** means the particle travels along the z-axis: it indicates that the magnetic field is directed along the z-axis. However, the particle's motion is not restricted to the z-axis alone. It has an initial velocity component in the y-direction, which means it will move in the x-y plane while also being influenced by the z-directed fields. The trajectory will be a helical path due to the combined effects of the electric and magnetic forces.

In summary, to find the minimum speed of the particle, use the relationship **v' = E' / B'**, and to determine the time when this occurs, analyze the motion in both the x and z directions, considering the forces acting on the particle. This will give you a comprehensive understanding of the particle's dynamics in the given fields.