Question

An electron is traveling horizontally from the northwest toward the southeast in a region of space where the Earth's magnetic field is directed horizontally toward the north. What is the direction of the magnetic force on the electron?

Short answer

Answer: The direction of the magnetic force on the electron is vertically downward.

Step-by-step solution

Identify the directions of velocity and magnetic field

The electron is moving horizontally from the northwest toward the southeast, so its velocity direction goes from northwest to southeast (let's call it the "v" vector). The Earth's magnetic field is directed horizontally toward the north (let's call it the "B" vector).

Apply the right-hand rule for positive charges

To apply the right-hand rule, point the fingers of your right hand in the direction of v-vector (from northwest to southeast). Then, curl your fingers in the direction of the Earth's magnetic field, B-vector (toward the north). Your thumb will point in the direction of the magnetic force acting on a positive charge, F-vector.

Reverse the direction for negative charges

Since the electron is a negatively charged particle, the direction of the force will be opposite to that determined in Step 2. To find the direction for a negatively charged particle, flip the direction of your thumb (180 degrees). This will give us the direction of the magnetic force acting on the electron.

Determine the direction of the magnetic force

After completing steps 2 and 3, we find that the direction of the magnetic force on the electron is vertically downward. This means that the magnetic force on the electron is pointing toward the ground. In conclusion, the direction of the magnetic force on the electron moving horizontally from northwest to southeast in the Earth's magnetic field directed horizontally toward the north is vertically downward.

Key concepts

Earth's magnetic field

Understanding Earth's magnetic field is crucial when exploring the magnetic force on electrons. Much like a bar magnet, Earth has a magnetic field that extends from the North Magnetic Pole to the South Magnetic Pole. Although invisible to the naked eye, this field plays a vital role in orienting compasses and affecting the behavior of charged particles, like electrons, within its influence.

Earth's field is not uniform and its intensity and direction vary with location. In general, the field resembles the one that would be generated by an enormous bar magnet tilted about 11 degrees from the Earth's rotational axis. For the exercise at hand, it is the horizontal component of the Earth's magnetic field that we are concerned with—specifically, the component that points horizontally toward the north.

Right-hand rule

When visualizing magnetic forces, the right-hand rule is an indispensable tool. It's a mnemonic that helps predict the direction of the magnetic force exerted on a moving charge in a magnetic field. To use it, extend your right hand with the fingers straight and the thumb at a right angle.

For positive charges, you align your fingers with the velocity vector (direction of charge movement), curl them toward the direction of the magnetic field, and your thumb will then point in the direction of the magnetic force. Remember, this initial step is for positive charges, so the rule must be modified for electrons, which carry a negative charge, by flipping the direction the force. This flipping is an essential aspect of the exercise improvement advice and facilitates a more profound comprehension of the charged particles' behavior in magnetic fields.

Velocity of charged particles

Velocity refers to the speed and direction of a moving object. For charged particles, like electrons, velocity is a significant determinant of their interaction with magnetic fields. In the context of our exercise, the electron has a specific diagonal trajectory from northwest to southeast. This aspect of direction is pivotal because the magnetic force experienced by a charge in a magnetic field is dependent not only on the speed but also on the direction of travel relative to the field.

The force's magnitude can be calculated by taking the product of the charge, its velocity, and the magnetic field's strength, and considering the angle between the direction of the charge's movement and the magnetic field. However, it's the direction of the velocity and its component perpendicular to the magnetic field that directly influences the direction of the magnetic force.

Negative charge behavior in magnetic fields

The behavior of negative charges, such as electrons, in magnetic fields differs from that of positive charges. While the force's magnitude on a charge moving through a magnetic field is the same for both positive and negative charges with the same velocity, the direction of the force is opposite due to their contrasting charges.

Recalling our electron in the exercise, we first deduced the force direction assuming a positive charge. To correct this for the electron's negative charge, we applied the right-hand rule and then reversed the direction of the force. This reversal is vital because it respects the fundamental principle that like charges repel and unlike charges attract, which governs the behavior of charged particles in magnetic fields. Following this principle, the negatively charged electron experiences a magnetic force in the exact opposite direction to that of a hypothetical positive charge moving in the same manner.