Respuesta :
Answer:
m/m' = 1/12
Explanation:
First, we need to equal the kinetic energy with the electric potential energy to determine the first ion's velocity:
[tex] \frac{1}{2} mv^{2} = q \Delta V [/tex] (1)
where m: first ion's mass, v: first ion's velocity, q: first ion's charge and ΔV: potential difference
[tex] v^{2} = \frac {2 q \Delta V}{m} [/tex]
[tex] v = \sqrt {\frac {2 q \Delta V}{m}} [/tex] (2)
To determine the mass m, we need to equal the centripetal force with the Lorentz force:
[tex] ma_{c} = qvB = m \frac{v^{2}}{R} [/tex]
where [tex] a_{c} [/tex]: centripetal acceleration = v²/R, B: magnetic field, R: first ion's semicircle radius
[tex] m = \frac {RqB}{v} [/tex] (3)
From the equations (2) and (3), we can get the final expression for the mass m:
[tex] m = \frac {RqB}{\sqrt {\frac {2 q \Delta V}{m}}} [/tex]
[tex] m = \frac{R^{2} qB^{2}}{2 \Delta V} [/tex] (4)
Similarly, for the second ion we have:
[tex] m' = \frac{R'^{2} q'B^{2}}{2 \Delta V} [/tex] (5)
Finally, from the equations (4) and (5), we can calculate the ratio of the masses of the ions:
[tex] \frac {m}{m'} = \frac{R^{2} q}{R'^{2} q'} = \frac{R^{2} q}{(2R)^{2} \cdot 3q} [/tex]
[tex] \frac {m}{m'} = \frac {1}{12} [/tex]
Have a nice day!
The ratio of the mass of the first and second particle is 1/12. The force exerted on the moving charged particle in electric and magnetic fields.
What is Lorentz Force?
It can be defined as the force exerted on the moving charged particle in electric and magnetic fields.
First, calculate the velocity of the particle, by equalizing kinetic and electric potential energy.
[tex]\dfrac 12 mv^2 = q\Delta V\\\\ v = \sqrt {\dfrac {2q\Delta V}{m}[/tex].............1
To find the mass equalize the Lorentz and centripetal force,
[tex]\dfrac {mv2}{r } = qvB\\\\ m = \dfrac {Rqb}{v} [/tex]..........2
Put the value of V in the 2nd equation,
[tex]m = \dfrac {Rqb}{ \sqrt {\dfrac {2q\Delta V}{m}}}\\ [/tex]
[tex] m = \dfrac {R^2qB^2}{ {2\Delta V}}\\ [/tex].............3
Since another mass has 3 times more charge with doubled radius,
So, the mass will be,
[tex]m'= \dfrac {2R'^23q'B^2}{ {2\Delta V}}\\[/tex]
Now, calculate the ratio of m to m',
[tex]\dfrac m{m'} = \dfrac {Rq}{2R^2 \times 3q}\\\\ \dfrac m{m'} = \dfrac 1{12}[/tex]
Therefore, the ratio of the mass of the first and second particle is 1/12.
Learn more about Lorentz Force:
https://brainly.com/question/13791875
