Question

A magnetic field is oriented in a certain direction in a horizontal plane. An electron moves in a certain direction in the horizontal plane. For this situation, there a) is one possible direction for the magnetic force on the electron. b) are two possible directions for the magnetic force on the electron. c) are infinite possible directions for the magnetic force on the electron.

Short answer

Answer: There are two possible directions for the magnetic force on the electron, which could be either into or out of the horizontal plane (upward and downward), depending on the relative orientation of the electron's velocity and the magnetic field.

Step-by-step solution

Recall Magnetic Force Equation

To find the magnetic force acting on an electron, we must recall the magnetic force equation, given by: F = q(v x B), where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field. The cross product (v x B) helps determine the direction of the magnetic force.

Apply Right-Hand Rule to Find Direction of Force

Using the right-hand rule, we can find the direction of the magnetic force on the electron. Point your fingers in the direction of the electron's velocity (v) and curl them in the direction of the magnetic field (B). Your thumb will then point in the direction of the magnetic force (F). Since the electron has a negative charge, the actual force will be in the opposite direction of your thumb.

Choose the Correct Option

Let's analyze the given options: a) There's only one possible direction for the magnetic force. This statement is not accurate because the force's direction depends on the relative orientation of both the electron's velocity and the magnetic field. b) There are two possible directions for the magnetic force. This option is true. By applying the right-hand rule, the magnetic force could be either into or out of the horizontal plane (upward and downward) because of the opposite direction due to the negative charge of the electron. c) There are infinite possible directions for the magnetic force. This statement is incorrect, as there are only two possible directions for the magnetic force acting on the electron based on the right-hand rule. So, the correct answer is option b) there are two possible directions for the magnetic force on the electron.

Key concepts

Magnetic Force Equation

Understanding the equation for the magnetic force acting on a particle such as an electron is fundamental in electromagnetism. The force is quantified by a straightforward but powerful formula: \( F = q(v \times B) \). In this equation, \( F \) represents the magnetic force, \( q \) is the charge of the particle, and \( v \) and \( B \) signify the velocity of the particle and the magnetic field, respectively. The operation \(v \times B \) is known as the cross product, indicating that the direction of the force is perpendicular to the plane formed by \(v\) and \( B\).

This equation shows that the magnitude of the force depends on the charge size, the particle's velocity, and the magnetic field's strength. Additionally, it implies that if either the velocity of the particle or the magnetic field, or both, are zero, then the magnetic force will also be zero. It's a vectors' interaction – one vector (the velocity) crossing with another (the magnetic field) to produce a third (the force), which is at a right angle to the plane of the first two.

Right-Hand Rule

The Right-Hand Rule is an easy-to-remember guide for predicting the direction of the magnetic force on a charged particle moving in a magnetic field. To apply this rule, follow these steps:

• Extend your right hand with your thumb and fingers straightened, and your palm facing up.
• Orient your fingers in the direction of the charged particle's velocity, \(v\).
• Rotate your hand so that your palm faces toward the direction of the magnetic field, \(B\).
• Your thumb now points in the direction of the force experienced by a positively charged particle.

Since an electron is negatively charged, the force direction will be opposite to the thumb. It's like telling the story of a journey – through the air (velocity), into the forest (magnetic field), and then the twist in the plot (the thumb pointing to the force on a positive particle). Now flip that tale upside-down for an electron, and you've mastered the Right-Hand Rule!

Electron Charge and Magnetic Fields

Diving deeper into how the charge of an electron interacts with magnetic fields reveals much about the behavior of subatomic particles in electromagnetic environments. An electron carries a fundamental unit of negative electric charge, denoted as \(-e\). This intrinsic property of the electron means that when it moves through a magnetic field, it experiences a force due to its charge.

However, it's not just any force — due to the charge being negative; the force direction is reversed compared to that experienced by a positive charge. This inverse relationship showcases the electron's dynamic ballet with the invisible tapestry of the magnetic fields around it. Just as dancers move to the rhythm, electrons twist and turn in response to the electromagnetic melody folded in space. This interplay results in a rich spectrum of physical phenomena, including electrical currents, magnetic storage of data, and the awe-inspiring auroras weaving colors through Earth's polar night skies.