Problem 74-5    Source:   Problem 25 - page 227 -    HALLIDAY, RESNICK, WALKER, Jearl -   Book: Fundamentos de Física - Vol 3 - Ed. LTC - 8ª edição - 2009.

A alpha particle, where q = +2 e and m = 4.00 u describes a circular path of 4.50 cm in radius in a region where there is a uniform magnetic induction field of modulus equal to B = 1.20 T. Determine:

a) the scalar velocity of the particle.

b) the period of revolution.

c) the kinetic energy of the alpha particle.

d) the potential difference necessary for the particle to reach the energy of item c).

Solution of the Problem 74-5   

Item a   

A particle alpha means a nucleus of the Helium atom, formed by two protons and two neutrons. For this reason it has a charge equivalent to the electric charge of two protons, that is, q = +2 e and, however its mass can be approximated by the mass of four protons, or m = 4 m_p. To determine the scalar velocity of the particle we will use eq. 74-05, reproduced below.

Making an algebraic arrangement to determine the value of v, we have:

Making the substitution by numeric values, we have:

Performing the calculation, we obtain:

Item b   

As we know the radius of the circumference and the speed we can use the laws of mechanics to find the period. Thus, the length of the circumference divided by the speed allows us to calculate the period. Soon, replacing with the numerical values, we have:

Performing the calculation, we obtain:

Item c   

We can calculate the energy of the alpha particle using the kinetic energy equation, studied in mechanics. In this way, knowing that the proton mass is worth m_p = 1.66 x 10 ^-27 Kg and, in addition, we have to m_α = 4 m_p. Thus, replacing with the numerical values:

Performing the calculation, we obtain:

When working with atomic particles it is common to express energy in eletron-volt. For this, the value found in joules is divided by the electron charge, which is worth q_e = 1,60 x 10^-19 C. So:

Item d   

The energy of the particle can also be calculated using the equation K = q ΔV. How do we want to calculate the potential difference so that the particle reaches the energy calculated in the previous item, we have: