Question

An electron is moving with a constant velocity. When it enters an electric field that is perpendicular to its velocity, the electron will follow a ________ trajectory. When the electron enters a magnetic field that is perpendicular to its velocity, it will follow a ________ trajectory.

Short answer

When an electron enters a perpendicular electric field, its trajectory is parabolic, whereas when it enters a perpendicular magnetic field, its trajectory is circular.

Step-by-step solution

1. Effect of electric field on electron's motion

When an electron enters an electric field, an electric force is exerted on it, which is given by F = qE, where q is the charge of the electron and E is the electric field strength. Since the electric field is perpendicular to the electron's initial velocity, the force will affect only the perpendicular component of its motion. The electric force will cause the electron to undergo uniform acceleration in the direction of the force. As a result, the trajectory of the electron will be parabolic.

2. Effect of magnetic field on electron's motion

When an electron enters a magnetic field, a magnetic force is exerted on it, which is given by F = qvBsinθ, where q is the charge of the electron, v is its velocity, B is the magnetic field strength, and θ is the angle between the velocity vector and the magnetic field vector. Since the magnetic field is perpendicular to the electron's initial velocity (θ=90°), the magnetic force will be F = qvB. The magnetic force acts perpendicular to both the velocity and the magnetic field direction, which means that the force and the magnetic field are always perpendicular to each other, resulting in a centripetal force that causes the electron to move in a circular trajectory.

3. Answer the question

When the electron enters an electric field that is perpendicular to its velocity, it will follow a parabolic trajectory. When the electron enters a magnetic field that is perpendicular to its velocity, it will follow a circular trajectory.

Key concepts

Electric Field Effects on Electron Trajectory

Electrons, those tiny negatively charged particles orbiting the nucleus of an atom, are significantly affected by electric fields. When an electron moves into an electric field that intersects perpendicularly with its initial direction of motion, things get interesting. The electric field exerts a force, and this force can be calculated with the equation \( F = qE \), where \( q \) is the electron charge and \( E \) is the electric field strength.

Because this force acts in a direction different from the electron's initial motion, the electron experiences uniform acceleration—meaning it speeds up at a steady rate in the direction of the electric force. This results in a trajectory that is parabolic. Just like a ball thrown into the air, the electron will move in a curved path, accelerating towards the direction of the force applied by the electric field.

Magnetic Field Effects on Electron Trajectory

Magnetic fields have a different effect on electrons compared to electric fields. When an electron enters a magnetic field perpendicularly to its velocity, it literally takes a turn. Here, the key force is the Lorentz force, described with \( F = qvBsin\theta \), where \( v \) is the velocity of the electron, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the direction of motion and the magnetic field—which is 90 degrees in our scenario.

At a right angle, \( \theta = 90^\circ \), hence \( sin\theta = 1 \). This simplifies the Lorentz force to \( F = qvB \). This force is always perpendicular to the direction of motion, acting as a centripetal force. Consequently, the electron, instead of accelerating in one direction, is continually pulled into a circular path. This motion is akin to a car turning in a track; the door pushes you towards the center—that's centripetal force keeping the electron in a circular trajectory.

Uniform Acceleration in Electric Fields

The concept of uniform acceleration is pivotal when discussing the trajectory of an electron in an electric field. Uniform acceleration implies a constant change of velocity; the speed of the electron changes at a consistent rate in a specific direction. The force exerted by the electric field on the electron (recall \( F = qE \)) provides this consistent shove.

The impact of this constant push is that the electron doesn't just move, it surges forward faster and faster along a set path—think of how an airplane gathers speed on a runway. This consistent acceleration aligns with the fundamental principle of physics that acceleration occurs when force is applied on a mass. Since the electron carries a charge, the electric field's influence directs this acceleration, morphing the trajectory into a predictable parabolic shape.

Centripetal Force in Magnetic Fields

Centripetal force is the hero behind those stunning loop-the-loop motions and all circular motion, guiding moving objects along a curved path. In the context of an electron moving through a magnetic field, this force doesn't speed the electron up or slow it down; instead, it keeps it moving round and round.

The magnetic force, which causes the centripetal force in this case, ensures that the electron stays on a circular path by constantly changing the direction of its velocity, not the magnitude. It's a bit like being on a merry-go-round, being pushed towards the center in order to maintain the circular ride. The consistent right-angle interaction between the velocity of the electron, the magnetic field, and the magnetic force helps create the perfect conditions for a centripetal force, and thus, perpetuates the electron's circular trajectory.