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Charged Particles in Magnetic Fields (SL IB Physics)

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Ann H

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Ann H

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Charged Particles in Magnetic Fields

  • When a charged particle enters a uniform magnetic field, it travels in a circular path
  • This is because the direction of the magnetic force F will always be
    • perpendicular to the particle's velocity v
    • directed towards the centre of the path, resulting in circular motion

Circular motion of charged particle, downloadable AS & A Level Physics revision notes

In a magnetic field, a charged particle travels in a circular path as the force, velocity and field are all perpendicular

  • The magnetic force F provides the centripetal force on the particle
  • The equation for centripetal force is:

F space equals space fraction numerator m v squared over denominator r end fraction

  • Equating this to the magnetic force on a moving charged particle gives the expression:

fraction numerator m v squared over denominator r end fraction space equals space B Q v

  • Rearranging for the radius r gives an expression for the radius of the path of a charged particle in a perpendicular magnetic field:

r space equals space fraction numerator m v over denominator B Q end fraction

  • Where:
    • r = radius of the path (m)
    • m = mass of the particle (kg)
    • v = linear velocity of the particle (m s−1)
    • B = magnetic field strength (T)
    • Q = charge of the particle (C)
  • This equation shows that:
    • Faster moving particles with speed v move in larger circles (larger r):  r space proportional to space v
    • Particles with greater mass m move in larger circles:  r space proportional to space m
    • Particles with greater charge q move in smaller circles:  r space proportional to space 1 over q
    • Particles moving in a strong magnetic field B move in smaller circles:  r space proportional to space 1 over B
  • The centripetal acceleration is in the same direction as the magnetic (centripetal) force
  • This can be found using Newton's second law:

F space equals space m a

Worked example

An electron travels 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 of the electron.

Answer:

Step 1: List the known quantities

  • Electron charge-to-mass ratio = e over m subscript e = 1.76 × 1011 C kg−1 (from formula sheet)
  • Magnetic flux density, B = 6.2 mT = 6.2 × 10−3 T
  • Speed of the electron, v = 3.0 × 106 m s−1

Step 2: Write an expression for the radius of an electron in a magnetic field

centripetal force = magnetic force

fraction numerator m subscript e v squared over denominator r end fraction space equals space B e v

r space equals space fraction numerator m subscript e v over denominator e B end fraction

Step 3: Substitute the known values into the expression

m subscript e over e space equals space fraction numerator 1 over denominator 1.76 cross times 10 to the power of 11 end fraction

r space equals space fraction numerator 3.0 cross times 10 to the power of 6 over denominator open parentheses 1.76 cross times 10 to the power of 11 close parentheses cross times open parentheses 6.2 cross times 10 to the power of negative 3 end exponent close parentheses end fraction space equals space 2.7 cross times 10 to the power of negative 3 end exponent space equals space 2.7 mm

Examiner Tip

Make sure you can derive the equation for the radius of the path of a particle travelling in a magnetic field.

As with orbits in a gravitational field, any object moving in circular motion will have a centripetal force and a centripetal acceleration. Make sure to refresh your knowledge of these equations.

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    Ann H

    Author: Ann H

    Expertise: Physics

    Ann obtained her Maths and Physics degree from the University of Bath before completing her PGCE in Science and Maths teaching. She spent ten years teaching Maths and Physics to wonderful students from all around the world whilst living in China, Ethiopia and Nepal. Now based in beautiful Devon she is thrilled to be creating awesome Physics resources to make Physics more accessible and understandable for all students no matter their schooling or background.