Question

A magnetic field \(B^{-}=\) Bo \(j \wedge\) exists in the region \(a

Short answer

The trajectory of the charge in the regions will be circular paths with opposite directions, making a figure-eight pattern, due to the positive and negative forces acting on the charge while it is moving in the magnetic field.

Step-by-step solution

Identify the Lorentz force formula in vector form

The Lorentz force acting on a particle with charge q and velocity \(\textbf{v}\) in a magnetic field \(\textbf{B}\) is given by the vector formula: \(\textbf{F} = q(\textbf{v} \times \textbf{B})\)

Calculate the force in each region

In the first region, with \(a < x < 2a\) and \(\textbf{B} = B_0j\): \(\textbf{F}_1 = q(\textbf{V}_0 \times B_0j) = q(V_0 \times B_0)k\) In the second region, with \(2a < x < 3a\) and \(\textbf{B} = - B_0j\): \(\textbf{F}_2 = q(\textbf{V}_0 \times (-B_0j)) = -q(V_0 \times B_0)k\) The force is in the z-direction in both regions, and the charge is positive, thus causing a force in a circular motion.

Find the equation of motion in each region

Starting with the first region, we can write the equation of motion using Newton's second law: \(m\frac{d^2 \textbf{r}}{dt^2} = q(V_0 \times B_0)k\) Next, we integrate the equation to find the position vector as a function of time: \(\int \frac{d^2\textbf{r}}{dt^2} dt = \frac{d\textbf{r}}{dt} = q(V_0\times B_0)t + C_1\) Integrating once more: \(\int \frac{d\textbf{r}}{dt} dt = \textbf{r} = \frac{q(V_0\times B_0)}{2}t^2 + C_1t + C_2\) Then, do the same for the second region with the negative force: \(m\frac{d^2 \textbf{r}}{dt^2} = -q(V_0 \times B_0)k\) Repeating the integration steps, we get the relation for the position vector of the second region: \(\textbf{r} = -\frac{q(V_0\times B_0)}{2}t^2 + C_3t + C_4\)

Analyze the trajectory

From the above equations, we can see that the motion is primarily in a circular path since the force is perpendicular to the velocity. In the first region, the path is a clockwise circle due to the positive force. In the second region, the path is an anti-clockwise circle due to the negative force. Therefore, the trajectory of the charge in the regions will be circular paths with opposite directions, making a figure-eight pattern.

Key concepts

Circular Motion

Circular motion occurs when an object moves along a curved path. In the context of a magnetic field, charged particles like electrons or positive ions often experience circular motion. This is due to the Lorentz force, which acts perpendicular to the velocity of the charge.

In our exercise, we explore how a positive charge moving through different regions of a magnetic field undergoes circular motion. The magnetic force causes this circular path because it constantly changes the direction of the velocity of the charged particle. However, it does not change its speed. This force acts as a centripetal force, which is essential for keeping the particle in circular motion.

For a charge in a magnetic field, the radius of the circle it moves in can be determined by the balance between the magnetic force and the centripetal force. This radius can be expressed using:

• the charge of the particle (\(q\)
• the velocity (\(v\)
• the magnetic field strength (\(B\)

The relation is given by \(r = \frac{mv}{qB}\). Understanding this concept helps explain why the charge moves along a circular path rather than a straight line.

Magnetic Field

A magnetic field exerts a force on moving electric charges and is represented by vector fields. In our exercise, the magnetic field changes in direction between the regions defined by \(a < x < 3a\).

In the first region, the magnetic field is positive (\(+B_0j\)), causing the force to be directed perpendicularly to the motion of the charge. This specific direction of force results in clockwise circular motion for the charge. In contrast, the second region has a negative magnetic field (\(-B_0j\)), which reverses the force's direction and makes the circular path anti-clockwise.

The presence and direction of a magnetic field are crucial in determining the motion of charged particles. The magnetic force is calculated using the cross-product of the velocity vector (\( extbf{v}\)) and the magnetic field vector (\( extbf{B}\)), which mathematically is represented as \( extbf{F} = q( extbf{v} \times extbf{B})\). This highlights how a magnetic field can be unique in terms of controlling trajectories and paths.

Charge Trajectory

The trajectory of a charged particle in a magnetic field depends heavily on the magnetic field's orientation and the charge's velocity. This exercise specifically highlights how a positive charge moves in distinct paths when it enters differently oriented magnetic fields.

As the charge enters each region, its trajectory is defined by the vector cross-product effect of its velocity and the magnetic field. This interaction creates a circular path for the particle. In the exercise, the regions cause the charge to move in opposite circular paths - clockwise in the first section and anti-clockwise in the second.

The result of these path interactions creates an interesting figure-eight pattern. The trajectory provides a clear example of how magnetic fields can influence charged particles significantly, deviating them from a linear path to a controlled path based on the circle's radius determined by equation \(r = \frac{mv}{qB}\). This concept is fundamental in comprehending the behavior of charged particles in varied magnetic fields.