Solved Examples: Finding Particle Radius in a Cyclotron

📚 Quick Study Guide

• 🧲 The magnetic force ($F_B$) acting on a charged particle moving with velocity ($v$) in a magnetic field ($B$) is given by: $F_B = qvB$, where $q$ is the charge of the particle.
• 🔄 In a cyclotron, this magnetic force provides the centripetal force ($F_c$) required for the circular motion: $F_c = \frac{mv^2}{r}$, where $m$ is the mass of the particle and $r$ is the radius of the circular path.
• ⚛️ By equating the magnetic force and the centripetal force, we can derive the formula for the radius: $qvB = \frac{mv^2}{r}$. Solving for $r$, we get $r = \frac{mv}{qB}$.
• ⚡️ The kinetic energy ($KE$) of the particle can be expressed as $KE = \frac{1}{2}mv^2$. If the kinetic energy is given, you can find the velocity $v = \sqrt{\frac{2KE}{m}}$.
• 💡 Remember to use consistent units: mass in kg, velocity in m/s, charge in Coulombs, magnetic field in Tesla, and radius in meters.

🧪 Practice Quiz

1. A proton (charge $1.6 \times 10^{-19}$ C, mass $1.67 \times 10^{-27}$ kg) moves in a cyclotron with a magnetic field of 1.0 T. If its velocity is $3 \times 10^7$ m/s, what is the radius of its path?
   1. 0.15 m
   2. 0.21 m
   3. 0.31 m
   4. 0.48 m
2. An alpha particle (charge $3.2 \times 10^{-19}$ C, mass $6.64 \times 10^{-27}$ kg) moves in a circular path of radius 0.20 m in a cyclotron with a magnetic field of 1.5 T. What is the velocity of the alpha particle?
   1. $1.5 \times 10^7$ m/s
   2. $2.0 \times 10^7$ m/s
   3. $3.0 \times 10^7$ m/s
   4. $4.5 \times 10^7$ m/s
3. An electron (charge $1.6 \times 10^{-19}$ C, mass $9.11 \times 10^{-31}$ kg) travels in a cyclotron with a magnetic field of 0.5 T. If the radius of its path is 0.05 m, what is its velocity?
   1. $2.2 \times 10^6$ m/s
   2. $4.4 \times 10^6$ m/s
   3. $6.6 \times 10^6$ m/s
   4. $8.8 \times 10^6$ m/s
4. A deuteron (charge $1.6 \times 10^{-19}$ C, mass $3.34 \times 10^{-27}$ kg) has a kinetic energy of $8.0 \times 10^{-13}$ J while moving in a cyclotron with a magnetic field of 2.0 T. What is the radius of its path?
   1. 0.10 m
   2. 0.16 m
   3. 0.20 m
   4. 0.26 m
5. A lithium ion (charge $1.6 \times 10^{-19}$ C, mass $1.16 \times 10^{-26}$ kg) moves in a cyclotron with a magnetic field of 0.8 T. If the radius of its path is 0.25 m, what is its kinetic energy?
   1. $4.6 \times 10^{-14}$ J
   2. $9.2 \times 10^{-14}$ J
   3. $1.8 \times 10^{-13}$ J
   4. $3.6 \times 10^{-13}$ J
6. A charged particle with a charge of $4.8 \times 10^{-19}$ C and a mass of $8.0 \times 10^{-27}$ kg moves in a cyclotron with a magnetic field of 1.2 T. If its kinetic energy is $6.0 \times 10^{-13}$ J, what is the radius of its circular path?
   1. 0.08 m
   2. 0.12 m
   3. 0.16 m
   4. 0.20 m
7. What happens to the radius of the path of a charged particle in a cyclotron if the magnetic field strength is doubled, assuming all other parameters remain constant?
   1. The radius is halved.
   2. The radius is doubled.
   3. The radius remains the same.
   4. The radius becomes four times larger.