Electrons pass through the parallel electrodes shown in FIGURE EX37.9 with a speed of 5.0×10⁶ m/s. What magnetic field strength and direction will allow the electrons to pass through without being deflected? Assume that the magnetic field is confined to the region between the electrodes.

Verified step by step guidance

1. Step 1: Understand the problem. Electrons are moving through a region with parallel electrodes, and we need to determine the magnetic field strength and direction that will allow the electrons to pass through without deflection. This means the magnetic force must counteract the electric force acting on the electrons.
2. Step 2: Recall the forces acting on a charged particle. The electric force is given by \( F_E = qE \), where \( q \) is the charge of the electron and \( E \) is the electric field strength. The magnetic force is given by \( F_B = qvB \), where \( v \) is the velocity of the electron and \( B \) is the magnetic field strength.
3. Step 3: Set the forces equal to each other for no deflection. Since the electron is not deflected, \( F_E = F_B \). Substituting the formulas, \( qE = qvB \). Simplify to find \( B \): \( B = \frac{E}{v} \).
4. Step 4: Determine the direction of the magnetic field. Use the right-hand rule for the magnetic force. The magnetic field must be perpendicular to both the velocity of the electrons and the electric field. If the electric field is vertical and the electrons are moving horizontally, the magnetic field will be directed into or out of the plane of the page.
5. Step 5: Calculate \( E \) if needed. If the electric field strength \( E \) is not provided, it can be determined using the voltage \( V \) across the electrodes and the distance \( d \) between them: \( E = \frac{V}{d} \). Substitute \( E \) and \( v \) into \( B = \frac{E}{v} \) to find the magnetic field strength.

Key Concepts

Lorentz Force

The Lorentz force describes the force experienced by a charged particle moving through a magnetic field. It is given by the equation F = q(v × B), where F is the force, q is the charge of the particle, v is its velocity, and B is the magnetic field. For an electron to pass through the electrodes without deflection, the magnetic force must balance any electric forces acting on it.

Magnetic Field Direction

The direction of the magnetic field is crucial in determining the behavior of charged particles. According to the right-hand rule, if you point your thumb in the direction of the electron's velocity and curl your fingers in the direction of the magnetic field, your palm will face the direction of the force on a positive charge. For electrons, which are negatively charged, the force will be in the opposite direction, influencing how the magnetic field must be oriented.

Velocity and Magnetic Field Relationship

The relationship between the velocity of charged particles and the magnetic field strength is essential for achieving no deflection. The condition for an electron to pass through without deflection is that the magnetic force equals the electric force acting on it. This requires a specific magnetic field strength, which can be calculated using the equation B = E/v, where E is the electric field strength and v is the velocity of the electrons.