An electron travels with speed 1.0 ×10^7 m / s between the two parallel charged plates shown in

step 1 Hi, there are two metallic charged plates kept parallel to each other. Upper plate is positively

An electron travels with speed 1.0 ×10^7 m / s between the...

An electron travels with speed $1.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ between the two parallel charged plates shown in Figure $\mathrm{P} 24.46 .$ The plates are separated by $1.0 \mathrm{cm}$ and are charged by a $200 \mathrm{V}$ battery. What magnetic field strength and direction will allow the electron to pass between the plates without being deflected?

Key Concepts

Electric Field Between Parallel Plates

A uniform electric field can be established between two parallel plates by applying a voltage difference across them. The strength of this field is determined by the relationship E = V/d, where V is the voltage and d is the separation between the plates. This concept is crucial for understanding how the initial force on the charged particle is generated.

Magnetic Field Calculation

Given the need to balance the electric force on a moving charged particle, the necessary magnetic field strength can be determined using the equation B = E/v. This direct relation is derived from the force balance condition, and is widely used in devices that rely on precise control of particle trajectories.

Lorentz Force

The Lorentz force is the total force experienced by a charged particle moving in electric and magnetic fields, defined by the equation F = q(E + v × B). This fundamental concept explains how both fields simultaneously affect the motion of the particle and is essential for analyzing trajectories in electromagnetic environments.

Balance of Forces in Crossed Electric and Magnetic Fields

When an electric field and a magnetic field are oriented such that their forces oppose each other, they can balance out, resulting in zero net force on the charged particle. The condition for this balance is given by E = vB (when considering the magnitude), ensuring that the electric force is exactly canceled by the magnetic force, allowing the particle to travel undeflected.

Determination of Magnetic Force Direction

The magnetic force on a moving charged particle is determined by the cross product v × B, which dictates that the force will be perpendicular to both the velocity of the particle and the magnetic field. Using the appropriate right-hand or left-hand rule (depending on the charge of the particle) is essential for predicting the correct deflection or cancellation of forces, thereby ensuring controlled movement through the fields.

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