Question

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true? (A) They will never come out of the magnetic field region. (B) They will come out travelling along parallel paths. (C) They will come out at the same time. (D) They will come out at different times.

Short answer

A is true; B, C, and D are false because the particles will move in circular paths within the magnetic field and will not exit the region unless an external force is applied.

Step-by-step solution

Understand the Motion of Charged Particles in a Magnetic Field

In a uniform magnetic field, a charged particle experiences a force that is perpendicular to both the velocity of the particle and the magnetic field. This force is given by Lorentz force equation, which is \( F = q(v \times B) \), where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field. An electron and a proton having the same velocity will experience forces of the same magnitude, but in opposite directions due to their opposite charges. This force will cause the particles to move in circular paths, with the radius of the path given by \( r = \frac{mv}{qB} \) where \( m \) is the mass of the particle.

Determine the Outcome for the Electron and Proton

Since the particles are moving in a perpendicular magnetic field, they will follow circular paths within the magnetic field region. Because the electron and proton have opposite charges, they will circulate in opposite directions. The radius of the circular path will be different for each, as the proton is much heavier than the electron.

Analyze the Given Statements

From the information in steps 1 and 2, you can now assess the given statements: (A) They will continue to move in circles within the magnetic field and thus will not leave the field region unless an external force acts upon them, making statement A true. (B) As they follow circular paths of different radii, they will not come out traveling along parallel paths, thus statement B is false. (C) They will not come out at all, which makes statement C false. (D) Since statement A is true and they never come out, the concept of 'coming out at different times' in statement D is irrelevant.

Key concepts

Lorentz Force

The Lorentz force is the fundamental mechanism explaining how charged particles interact with magnetic fields. It's the force experienced by a moving charged particle in the presence of electric and magnetic fields. This force is pivotal in the study of electromagnetism and can be quantified by the formula \( F = q(v \times B) \), where \( F \) is the force, \( q \) is the charge of the particle, \( v \) is the velocity of the particle, and \( B \) is the magnetic field intensity.

In the context of our exercise, the Lorentz force points out why an electron and a proton, upon entering a magnetic field, do not continue travelling in straight paths. The force results in a change in their direction of motion but interestingly doesn't affect their speed since it acts perpendicular to the velocity.

Circular Motion of Charged Particles

When a charged particle like an electron or proton moves through a magnetic field in such a way that its path is perpendicular to the field lines, it begins to traverse a circular path. This is due to the Lorentz force acting as a centripetal force, which continuously changes the direction of the particle's velocity. The radius of this circular path is given by the equation \( r = \frac{mv}{qB} \), where \( m \) represents the mass of the particle, reinforcing the idea that the mass of the particle has a direct impact on the path's radius.

Students often wonder how the same magnetic field can result in different circular trajectories for different particles. The key lies in the mass and charge of the particles: since the electron and the proton have different masses, their circular paths differ in size, influencing their motion within the magnetic field.

Magnetic Field Effects on Particles

The magnetic field does not merely alter the path of a charged particle; it fundamentally changes its motion without altering its linear momentum or kinetic energy. The particle's velocity remains constant in magnitude because the magnetic field does work only to change the direction of the velocity, not its magnitude.

For our exercise, this indicates that while electrons and protons enter the magnetic field region with the same velocity, their trajectories bend into circular motions due to the Lorentz force. They stay within the magnetic field because the force required to remove them from their circular path isn't provided by the field itself. This reinforces why they can't exit the magnetic region under these set conditions.

Comparative Motion of Electron and Proton

In our scenario, both an electron and a proton enter a magnetic field with the same velocity but due to their mass and charge differences, their behavior inside the field is notably different. Electrons, being much less massive than protons, will have a much smaller radius of circular motion—implying a quicker, tighter spiral. On the contrary, protons with greater mass move in larger circles and take more time to complete each loop.

This comparative aspect of motion is crucial for understanding the exercise scenarios. Even though they both experience forces of equal magnitude, the direction and pathway differ, leading to distinct motions within the magnetic field. The fundamental differences in their properties mean that electrons and protons can never re-emerge from the magnetic field at the same time or along parallel paths, debunking the suggested options in the initial exercise question.