   Chapter 19, Problem 20P

Chapter
Section
Textbook Problem

A proton (charge +e, mass mp), a deuteron (charge +e, mass 2mp), and an alpha particle (charge +2e, mass 4mp) are accelerated from rest through a common potential difference ΔV. Each of the particles enters a uniform magnetic field B → , with its velocity in a direction perpendicular to B → . The proton moves in a circular path of radius τp. In terms of τp, determine (a) the radius rd of the circular orbit for the deuteron and (b) the radius rα for the alpha particle.

a)

To determine
The radius rd of the circular path for the deuteron in terms of the radius of proton path rp .

Explanation

Given info: A proton (charge +e, mass mp ), a deuteron (charge +e, mass 2mp ), and an alpha particle (charge + 2e, mass 4mp ) areaccelerated from rest through common potential difference ΔV . The magnetic field is B . The particles velocities and the magnetic field are perpendicular to each other. The radius of the protons path is rp . The deuteron of the protons path is rd . The radius of the alpha particles path is rα .

Explanation:

Consider a particle which is accelerated from rest through a potential difference ΔV . Charge of the particle is q and mass is m .

From conservation of energy,

KEf+PEf=KEi+PEi

Since the initially the particle is at rest,

KEf+PEf=0+PEi

On re-arrangement,

KEf=PEiPEf       (1)

Since the potential energy of a charged particle is charge times potential,

12mv2=qViqVf=q(ViVf)=qΔV

On re-arrangement,

v=2qΔVm       (2)

Since the magnetic field supplies the centripetal acceleration,

qv

b)

To determine
The radius rα of the circular path for the alpha particle in terms of the radius of proton path rp .

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