Charged Particles in Electric & Magnetic Fields

A charged particle moving at a constant speed in electric and magnetic fields orientated at right angles to each other will continue to travel at a constant speed

One particular orientation is:
- Motion of particle, v to the right
- Magnetic field B coming out of the page on the z-axis
- Electric field E going up the page on the y-axis

Hence, the three vectors are perpendicular to each other

4-2-perpendicular-electric-and-magnetic-fields

An example of the orientation of an electric field perpendicular to a magnetic field

Situation 1: Moving Positively Charged Particle

The electric force is parallel to the electric field, so it acts up the page
The magnetic force is perpendicular to the magnetic field according to Fleming's left hand rule
When applied to the orientation above:
- The positive charge moves to the right, this is the second finger
- The magnetic field comes out of the page, this is the first finger
- So, the magnetic force is acting down the page, this is the thumb

Hence, the electric force and magnetic force act in opposite directions on the charge

4-2-perpendicular-electric-and-magnetic-fields-with-forces

The electric force acts up the page and the magnetic force acts in the opposite direction down the page

Situation 2: Moving Negatively Charged Particle

The charge is negative, so the electric force acts in the opposite direction to the electric field, down the page
The magnetic force is perpendicular to the magnetic field according to the Fleming's left hand rule
When applied to the orientation above:
- A positive charge moves to the left, this is the second finger (since the negative moves to the right, in the opposite direction)
- The magnetic field comes out of the page, this is the first finger
- So, the magnetic force is acting up the page, this is the thumb

4-2-perpendicular-electric-and-magnetic-fields-with-forces-negative

The magnetic force acts up the page and the electric force acts in the opposite direction down the page

Strength of Forces

Adjusting the strengths of the two fields the magnitude of the electric and magnetic forces can be made equal
So the forces cancel each other out
Hence, the particle continues its original motion in a straight line with its constant speed v

Calculating Particle Velocity

When the two forces are equal then electric force, F_E = magnetic force, F_B

F_E = Eq and F_B = Bqv

So, Eq = Bqv

and Bv = E

Hence, $v = \frac{E}{B}$

The equation for the magnetic force F_B on charge is introduced in this revision note

Worked example

An electron passes between two parallel metal plates moving with a constant velocity of 2.1 × 10⁷ m s⁻¹. The potential difference between the plates is 3100 V. A uniform magnetic field of magnitude 0.054 T acts perpendicular to the electric field and the movement of the electron.

The electric field acts to the right and the electron is moving downwards.

(a)

Determine the direction of the magnetic field

(b)

Calculate the separation of the plates

Answer:

(a) The direction of the magnetic field:

Step 1: Draw a diagram of the situation

4-3-3-worked-example-ma

The electric field goes (from the positive plate to the negative plate), to the right
The electron is moving vertically downwards
So, the current is moving upwards in the opposite direction to the electron
The electric force is acting in the opposite direction to the electric field because the particle is an electron

Step 2: Determine the direction of the magnetic field

The electron is moving at a constant speed, so the magnetic and electric forces are equal and opposite
- Hence, the magnetic force acts to the left

(b) Calculate the separation of the plates:

Step 1: Calculate the magnitude of the electric field, E

$v = \frac{E}{B} \Rightarrow E = v B$

$E = (2.1 \times 10^{7}) \times 0.054 = 1.134 \times 10^{6} N C^{- 1}$

Step 2: Calculate the separation of the plates

Use the electric field strength equation:

$E = \frac{V}{d} \Rightarrow d = \frac{V}{E}$

$d = \frac{3100}{1.134 \times 10^{6}}$

$d = 2.73 \times 10^{- 3}$ m

Exam Tip

Take time to consider the direction of all components of the electric and magnetic fields.

Remember that the electric and magnetic forces act in the opposite direction for negatively charged particles compared to positively charged.

The direction of the charge in Fleming's left hand rule is always the direction of positive charge. This should be in the opposite direction if the particle has a negative charge!

Charged Particles in Electric & Magnetic Fields (SL IB Physics)

Revision Note