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

Answer: When an electron enters an electric field that is perpendicular to its velocity, it follows a parabolic trajectory. And when entering a magnetic field that is perpendicular to its velocity, it follows a circular trajectory.

Step-by-step solution

Electric Field Definition

An electric field is a region in space where an electric charge experiences a force due to the presence of other charged particles. The force acting on a charged particle in an electric field depends on the charge of the particle and the strength of the electric field.

Magnetic Field Definition

A magnetic field is a region in space where a moving charge (or magnetic material) experiences a force due to the presence of other moving charges (or magnets). The force acting on a moving charge in a magnetic field depends on the charge, its velocity, and the strength of the magnetic field.

Electric field force on electron

When an electron enters an electric field, it experiences a force given by the product of its charge (negative) and the electric field (\(qE\)), where q is the charge of the electron and E is the electric field vector. Since the force is perpendicular to the initial velocity, the electron will experience a centripetal acceleration, causing it to move in a parabolic path.

Magnetic field force on electron

When the electron enters a magnetic field, it experiences a force given by the cross product of its velocity vector and the magnetic field vector (\(qvBsin\theta\)), where q is the charge of the electron, v is its velocity, B is the magnetic field vector, and \(\theta\) is the angle between the velocity and magnetic field vectors. Assuming the magnetic field is perpendicular to the velocity (\(\theta=90^\circ\)), the force acting on the electron is always perpendicular to its velocity, which means no work is done on the particle. Therefore, the electron will have a constant speed and move in a circular path. Answer: For the given blanks in the exercise, When an electron enters an electric field that is perpendicular to its velocity, it follows a __parabolic__ trajectory. And when entering a magnetic field that is perpendicular to its velocity, it follows a __circular__ trajectory.

Key concepts

Electric Field

The electric field is a pivotal concept in physics, describing how electric charges influence each other. It is an invisible force field that exists around all electrically charged objects, affecting the behavior of charges within its reach. Specifically, when an electron—a negative charge—moves into an electric field that is perpendicular to its motion, it is acted upon by an electric force that causes an alteration to its path.

The force exerted on a charge in an electric field is calculated with the formula \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) signifies the electric field strength. Since the electric force applies continuously at a right angle to the electron's motion, it doesn't speed up or slow down the electron but instead causes it to curve, tracing a parabolic trajectory. This is similar to the effect gravity has on projectiles on Earth, bending their paths into curves.

Magnetic Field

In the realm of physics, magnetic fields are as fundamental as electric fields but operate under slightly different principles. A magnetic field is generated by moving electric charges, like electrons in a wire, or by magnetic materials such as iron. Inside a magnetic field, a moving charge experiences a magnetic force, which is determined by the Lorentz force law.

The formula for the magnetic force is \( F = qvBsin\theta \), where \( F \) stands for force, \( q \) the charge, \( v \) the velocity, \( B \) the magnitude of the magnetic field, and \( \theta \) the angle between the velocity and magnetic field. When \( \theta \) is 90 degrees, meaning the magnetic field is perpendicular to the electron's motion, it follows a circular trajectory because the force is always directed towards the center of this circular path—creating centripetal acceleration—yet it doesn't alter the electron's speed.

Centripetal Acceleration

Centripetal acceleration is the core force in circular motion, always pointing towards the center of the rotation. It's the reason anything moving in a circular path doesn't go flying off in a straight line due to inertia. Instead, some force—like the magnetic force on an electron—keeps pulling it inward.

Mathematically, centripetal acceleration \( a_c \) is defined as \( a_c = \frac{v^2}{r} \), where \( v \) is the velocity of the moving object and \( r \) is the radius of the circular path. When an electron travels in a magnetic field perpendicularly to its velocity, the magnetic force provides the required centripetal force to sustain the circular motion, maintaining the electron's constant speed along its curvilinear path.

Circular Motion

Circular motion is a movement of an object along the circumference of a circle or another circular path. It's a classic representative of the concept of centripetal acceleration, where an inward force is necessary for keeping the object in motion along a curved path rather than allowing it to move off in a straight line due to inertial tendencies.

For the electron moving in a magnetic field, this inward force—arising due to magnetic interactions—ensures a stable circular path. Since the force is perpendicular to the electron's velocity and doesn't do any work (energy remains constant), the speed of the electron stays the same, although its direction continually changes to keep it on a circular trajectory.

Parabolic Trajectory

A parabolic trajectory is a two-dimensional, symmetrical curve described by an object moving under the influence of a constant force at a right angle to its initial velocity—often gravity for projectiles on Earth, or an electric field for charged particles like electrons. This trajectory—parabolic in nature—results from the constant acceleration due to the force, which changes the object's direction but not its speed along the axis perpendicular to the force.

In the case of an electron in an electric field, the force exerted causes the electron to move in a controlled, predictable manner, curving its path into the shape of a parabola. Understanding this motion helps us to predict the behavior of charges in various electrical and magnetic applications, from television tubes to particle accelerators.