Motion of charged particles (6.3.2)
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When a charged particle is in motion within a uniform magnetic field, the particle experiences a magnetic force of magnitude:
Where:
- if the magnetic force in newtons ,
- is the magnetic flux denity in teslas ,
- is the charge of the particle expressed in coulombs ,
- is the speed of the particle in metres per second , and
- is the angle between the particle’s direction of motion and the magnetic field direction.
For an electron or a proton, the charge is .
When the magnetic field lines and the direction of motion of the charge are perpendicular, the magnetic force is maximal and is equal to:
When the magnetic field lines and the direction of motion of the charge are parallel, the magnetic force is equal to zero:
Question walkthrough
Force on a Charged Particle in a Field
Calculate the magnetic force on a moving electron using F = QvB, given the field strength, particle speed, and perpendicular field orientation.
A particle of charge travelling in a magnetic field experiences a magnetic force .
When the velocity vector of the particle is perpendicular to the magnetic field vector , the magnetic force acts on the particle perpendicularly to both and .
It is important to note that the particle will follow a circular trajectory because the magnetic force is perpendicular to the velocity. The magnetic force will always point toward the centre of the circle, acting as a centripetal force.
A variation of Fleming’s left-hand rule can be used on charged particles to determine the direction of the magnetic force acting on that particle.
For a positive charge, the index finger of the left hand points in the direction of the magnetic field, the middle finger in the direction of motion, and the thumb points in the direction of the magnetic force.
Follow the same process for a negative charge, but flip the direction of the magnetic force at the end.
It is important to recall and understand the proper use of Fleming’s left-hand rule and its variation.
The magnetic force, , is a centripetal force that acts on the charged particle. Hence, we can apply the formula for a centripetal force:
However, centripetal acceleration can be expressed in terms of speed, , and radius, , of circular motion as:
Finally, combining the previous two equations, we get:
Where:
- is the mass of the particle in ,
- its speed in and
- is the radius of the circle traversed by the particle, in .
Knowing that the magnetic force is also equal to:
The equation becomes:
Thus, the radius of the circle completed by the charged particle is:
The radius of the circle can be increased by:
- increasing the mass of the charge ,
- increasing the speed of the charge ,
- reducing the magnetic flux density , and
- reducing the quantity of charge .
Question walkthrough
Finding Radius of a Proton's Circular Path in a Field
Calculate the radius of the circular path of a proton moving at right angles to a uniform magnetic field, given its charge-to-mass ratio and speed.
A velocity selector is a device used to separate out charged particles moving at a specific velocity from a stream of charged particles with a mixture of velocities.
Velocity selectors are used in mass spectrometers to produce beams of charged particles all moving at the same velocity.
A velocity selector consists of two parallel, oppositely charged conductors that create an electric field emanating from the positive plate to the negative plate.
A magnetic field of magnetic flux density is applied perpendicularly to the electric field lines. The direction of the magnetic field is chosen depending on the sign of the charged particle.
When a positively charged particle enters a velocity selector with speed , the particle experiences a rightward electric force equal to . Simultaneously, the positively charged particle experiences a leftward magnetic force equal to .
If the forces and are not balanced, the particle will deflect sideways.
When the forces are balanced, the particle will leave the velocity selector without deviation. When the electric and magnetic forces are balanced:
Hence, the velocity of the charged particle that continues without deviation in a velocity selector can be determined using the expression:
We can control which particle passes through the velocity selection region without deflection by varying the electric field strength and the magnetic flux density. The rest of the particles moving at different speeds will strike the sides of the velocity selector and, therefore, be prevented from exiting.
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Analysing Forces and Motion in a Velocity Selector
Determine the force directions, magnetic flux density, and subsequent particle motion in a velocity selector using the electric field strength and known exit speed.