Question

Which particles will have minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field
A . ${Li^+}$
B . Electron
C . Proton
D . ${He^+}$

Answer 1

Hint: In this question we have to find the frequency of revolution of lithium ions, an electron, a proton and a helium ion when projected with the same velocity perpendicular to a magnetic field. There is a centripetal force operating on a charged particle that revolves in a circular path in a uniform magnetic field, and that force is equal to the magnetic force acting on the particle.


Formula used:
Magnetic force on the particle:
${F_m}$= q(v×B); here, q denotes the charge, v is the velocity of the particle, and B the magnetic field.
The centripetal force is given by,
${{F}_{c}}=\dfrac{m{{v}^{2}}}{r}$
Here m is the mass of the particle, v is the velocity of the particle, and r is the radius.
The expression for frequency:
$f=\dfrac{v}{2\pi r}$; here, v is the velocity of the particle and r is the radius.



Complete answer:
Magnetic force on the particle:
${F_m}$= q(v×B); here, q denotes the charge, v is the velocity of the particle, and B the magnetic field.
The centripetal force is given by,
${{F}_{c}}=\dfrac{m{{v}^{2}}}{r}$
Here m is the mass of the particle, v is the velocity of the particle, and r is the radius.
The centripetal force counterbalances the magnetic pull that acts on a moving particle perpendicular to its velocity. The equilibrium formula states the following:
${F_m}$(magnetic force)=${F_c}$(centripetal force) - (i)
$qvB=\dfrac{m{{v}^{2}}}{r}$
$\dfrac{v}{r}=\dfrac{qB}{m}$
We know that the formula for frequency is:
$f=\dfrac{v}{2\pi r}=\dfrac{qB}{2\pi m}$
The frequency of a particle depends on its mass, charge and the magnetic field.
The value of the magnetic field and charge is the same for all the particles. Therefore, the frequency depends on the mass of the particle inversely.
$f\alpha \dfrac{1}{m}$
${F_{minimum}}$ will occur for a particle with maximum mass.
Among the given particles, ${Li^+}$ have the maximum mass and thus will have the minimum frequency.
The correct answer is A.




Note: ${Li^+}$ ion has 3 protons and 4 neutrons and ${He^+}$ ion has 2 protons and 2 neutrons making ${Li^+}$ ion the heaviest.
When a charged particle of charge 'q' and mass 'm' is projected to a magnetic field perpendicular to the magnetic field then it revolves with a certain frequency which depends on the mass and charge of the particle and the magnetic field.