An electron in a cathode-ray beam passes between 2.5-cm-long parallel-plate electrodes that are 5.0 mm apart. A 2.0 mT, 2.5-cm-wide magnetic field is perpendicular to the electric field between the plates. The electron passes through the electrodes without being deflected if the potential difference between the plates is 600 V. What is the electron's speed?

Verified step by step guidance

1. Identify the forces acting on the electron: The electron is subject to both an electric force due to the electric field between the plates and a magnetic force due to the magnetic field. For the electron to pass through without deflection, these forces must balance each other.
2. Write the expression for the electric force: The electric force \( F_E \) is given by \( F_E = eE \), where \( e \) is the charge of the electron and \( E \) is the electric field. The electric field \( E \) can be calculated using \( E = \frac{V}{d} \), where \( V \) is the potential difference (600 V) and \( d \) is the separation between the plates (5.0 mm or 0.005 m).
3. Write the expression for the magnetic force: The magnetic force \( F_B \) is given by \( F_B = e v B \), where \( v \) is the speed of the electron and \( B \) is the magnetic field strength (2.0 mT or 0.002 T).
4. Set the forces equal to each other: Since the electron is not deflected, the electric force and magnetic force must balance. Set \( F_E = F_B \), which gives \( eE = e v B \). Simplify this to \( E = v B \).
5. Solve for the electron's speed: Substitute \( E = \frac{V}{d} \) into \( E = v B \), giving \( \frac{V}{d} = v B \). Rearrange to solve for \( v \): \( v = \frac{V}{d B} \). Substitute the given values for \( V \), \( d \), and \( B \) to calculate the speed of the electron.

Key Concepts

Electric Field

An electric field is a region around charged particles where other charged particles experience a force. It is defined by the voltage difference between two points, which creates a force on charged particles like electrons. The strength of the electric field (E) can be calculated using the formula E = V/d, where V is the potential difference and d is the distance between the plates.

Magnetic Field

A magnetic field is a vector field that exerts a force on moving charged particles, such as electrons. The force experienced by a charged particle in a magnetic field is given by the Lorentz force law, F = q(v × B), where q is the charge, v is the velocity of the particle, and B is the magnetic field strength. The direction of the force is perpendicular to both the velocity of the particle and the magnetic field.

Velocity of Charged Particles

The velocity of charged particles, like electrons, is crucial for understanding their motion in electric and magnetic fields. In this scenario, the electron's speed can be determined by balancing the electric force and the magnetic force acting on it. When these forces are equal and opposite, the particle moves in a straight line without deflection, allowing us to calculate its speed using the relationship between the electric field, magnetic field, and the forces involved.