(a) What value of magnetic field would make a beam of...

(a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of $4.8 \times 10^{6} \mathrm{m} / \mathrm{s}$ , go undeflected through a region where there is a uniform electric field of $10,000 \mathrm{V} / \mathrm{m}$ pointing vertically up? ( $b$ ) What is the direction of the magnetic field if it is known to be perpendicular to the electric field? $(c)$ What is the frequency of the circular orbit of the electrons if the electric field is turned off?

Key Concepts

Right-Hand Rule and Charge Sign Consideration

The right-hand rule is used to determine the direction of the magnetic component of the Lorentz force. However, because electrons have a negative charge, the actual direction of the force is opposite to that indicated by the right-hand rule. Understanding this inversion is vital for correctly predicting the direction of the magnetic field required to counteract an electric field-induced force.

Cyclotron Frequency

In a situation where a charged particle moves in a magnetic field with no electric field present, it will undergo circular motion. The frequency of this motion, called the cyclotron frequency, is determined by the charge-to-mass ratio of the particle and the magnetic field strength. The formula for cyclotron frequency is derived from equating the magnetic force to the centripetal force required for circular motion.

Lorentz Force

The Lorentz force law is the foundational principle that governs the motion of charged particles in electric and magnetic fields. It states that the total force on a charge is the vector sum of the electric force, which is proportional to the electric field, and the magnetic force, which is proportional to the cross product of the particle’s velocity and the magnetic field.

Force Cancellation in Uniform Fields

When both electric and magnetic fields are present, it is possible to adjust the fields such that their respective forces on a charged particle cancel each other. This requires the magnetic force (which depends on velocity and field strength) to be equal in magnitude and opposite in direction to the electric force. This concept is often used to allow charged particles to travel undeflected in the combined fields.

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