A 150-V battery is connected across two parallel metal...

A $150-\mathrm{V}$ battery is connected across two parallel metal plates of area 28.5 $\mathrm{cm}^{2}$ and separation 8.20 $\mathrm{mm} .$ A beam of alpha particles (charge $+2 e,$ mass $6.64 \times 10^{-27} \mathrm{kg} )$ is accelerated from rest through a potential difference of 1.75 $\mathrm{kV}$ and enters the region between the plates perpendicular to the electric field, as shown in Fig. $\mathrm{E} 27.31 .$ What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

Key Concepts

Right-Hand Rule for Magnetic Force Direction

The right-hand rule is used to determine the direction of the magnetic force on a moving charged particle. By arranging the right hand so that the fingers point in the direction of the particle's velocity and then curling them toward the magnetic field, the thumb will point in the direction of the force, which is essential for aligning fields to achieve zero net deflection.

Velocity Selector Principle

For a particle to pass through a region with perpendicular electric and magnetic fields without deflection, the electric force (qE) must exactly cancel the magnetic force (qvB). This balance leads to the condition E = vB, allowing for the determination of the required magnetic field when the particle's speed is known.

Conservation of Energy and Kinetic Energy of Charged Particles

When a charged particle is accelerated from rest by a potential difference, its kinetic energy increase equals the charge multiplied by the voltage. This principle is used to calculate the velocity of the particle after acceleration using the formula ½mv² = qV.

Electric Field Between Parallel Plates

In a parallel plate arrangement, the electric field is uniform and can be determined by dividing the potential difference by the separation distance between the plates. This field exerts a force on any charged particle within the region, given by F = qE.

Lorentz Force and Magnetic Force

A charged particle moving in a magnetic field experiences a force called the Lorentz force, which acts perpendicular to both its velocity and the magnetic field. The magnitude of this force is determined by F = qvB when the velocity is perpendicular to the field.

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