# 4.35 Radius of a Charged Particle in a Magnetic Field

## Radius of a Charged Particle in a Magnetic Field

• A charged particle in uniform magnetic field which is perpendicular to its direction of motion travels in a circular path
• This is because the magnetic force F is always perpendicular to its velocity v
• F will always be directed towards the centre of orbit A charged particle travels in a circular path in a magnetic field

• The magnetic force F provides the centripetal force on the particle
• The equation for centripetal force is: • Where:
• F = centripetal force (N)
• m = mass of the particle (kg)
• v = linear velocity of the particle (m s1)
• r = radius of orbit (m)

• Equating this to the magnetic force on a moving charged particle gives the equation: • Rearranging for the radius r obtains the equation for the radius of the orbit of a charged particle in a perpendicular magnetic field: • The product of mass m and velocity v is momentum p
• Therefore, the radius of the charged particle in a magnetic field can also be written as: • Where:
• r = radius of orbit (m)
• p = momentum of charged particle (kg m s–1)
• B = magnetic field strength (T)
• q = charge of particle (C)

• This equation shows that:
• Particles with a larger momentum (either larger mass m or speed v) move in larger circles, since r ∝ p
• Particles with greater charge q move in smaller circles: r 1 / q
• Particles moving in a strong magnetic field B move in smaller circles: r 1 / B

#### Worked example

An electron with charge-to-mass ratio of 1.8 × 1011 C kg-1 is travelling at right angles to a uniform magnetic field of flux density 6.2 mT. The speed of the electron is 3.0 × 106 m s-1.

Calculate the radius of the circular path travelled by the electron. #### Exam Tip

Make sure you're comfortable with deriving the equation for the radius of the path of a charged particle travelling in a magnetic field, as this is a common exam question.

Crucially, the magnetic force is always perpendicular to the velocity of a charged particle. Hence, it is a centripetal force and the equations for circular motion can be applied. ### Get unlimited access

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