Charged Particles in Magnetic Fields
- 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 FB will always be perpendicular to its velocity v
- FB will always be directed towards the centre of the path
A charged particle moves travels in a circular path in a magnetic field
- The magnetic force FB provides the centripetal force on the particle
- Recall the equation for centripetal force:
- Where:
- m = mass of the particle (kg)
- v = linear velocity of the particle (m s-1)
- r = radius of the orbit (m)
- Equating this to the 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:
- This equation shows that:
- Faster moving particles with speed v move in larger circles (larger r): r ∝ v
- Particles with greater mass m move in larger circles: r ∝ m
- 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 circle path of the electron.
Answer:
Step 1: Write down the known quantities
- Charge-to-mass ratio,
= 1.8 × 1011 C kg–1 - Magnetic flux density, B = 6.2 mT
- Electron speed, v = 3.0 × 106 m s-1
Step 2: Write down the equation for the radius of a charged particle in a perpendicular magnetic field
Step 3: Substitute in values
Flemming's Left-Hand Rule
- The direction of the force on a charge moving in a magnetic field is determined by the direction of the magnetic field and the current
- Recall that the direction of the current is in the direction of conventional current flow (positive to negative)
- When the force, the magnetic field and the current are all mutually perpendicular to each other, the directions of each can be interpreted by Fleming’s left-hand rule
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- The thumb points in the direction of motion of the rod (or the direction of the force) (F)
- The first finger points in the direction of the external magnetic field (B)
- The second finger points in the direction of conventional current flow (I)On the left hand, with the thumb pointed upwards, first finger forwards and the second finger to the right, ie. all three are perpendicular to each other
Fleming’s left-hand rule
- Since this is represented in 3D space, sometimes the current, force or magnetic field could be directed into or out of the page, not just left, right, up and down
- The direction of the magnetic field into or out of the page in 3D is represented by the following symbols:
- Dots (sometimes with a circle around them) represent the magnetic field directed out of the plane of the page
- Crosses represent the magnetic field directed into the plane of the page
The magnetic field into or out of the page is represented by circles with dots or crosses
- The way to remember this is by imagining an arrow used in archery or darts:
- If the arrow is approaching head-on, such as out of a page, only the very tip of the arrow can be seen (a dot)
- When the arrow is receding away, such as into a page, only the cross of the feathers at the back can be seen (a cross)
Worked example
State the direction of the current flowing in the wire in the diagram below.
Answer:
- Using Fleming’s left-hand rule:
- B = into the page
- F = vertically downwards
- I = from right to left
Exam Tip
Don’t be afraid to use Fleming’s left-hand rule during an exam. Although, it is best to do it subtly in order not to give the answer away to other students!
