Welcome to Deflection of Charged Particles!
Have you ever wondered how old-style television screens painted glowing pictures, how particle accelerators like the Large Hadron Collider steer particles near the speed of light, or how scientists measure the exact mass of individual molecules in a sample? The answer lies in how we control moving charged particles using electric and magnetic fields.
Don't worry if this topic sounds intimidating at first! We are simply going to build upon ideas you already know: forces, Newton's laws, and projectile motion. Let's break it down step by step.
1. Charged Particles in a Uniform Electric Field
When a particle with charge \(q\) enters a region containing a uniform electric field of strength \(E\), it experiences a constant electrostatic force \(F_E\).
The Electric Force
The magnitude of the force is given by the formula:
\(F_E = Eq\)
Where:
• \(F_E\) = electrostatic force on the particle (in newtons, \(\text{N}\))
• \(E\) = electric field strength (in volts per metre, \(\text{V m}^{-1}\), or newtons per coulomb, \(\text{N C}^{-1}\))
• \(q\) = charge of the particle (in coulombs, \(\text{C}\))
Direction of Force:
• A positive charge feels a force in the same direction as the electric field lines.
• A negative charge (such as an electron) feels a force in the opposite direction to the electric field lines.
Trajectory: Parabolic Motion (The Gravity Analogy)
Imagine throwing a ball horizontally off a cliff. Gravity pulls it downward with a constant acceleration while its horizontal speed stays constant, tracing out a parabolic path.
The exact same thing happens when a charged particle enters a uniform electric field at right angles (perpendicular) to the field lines:
• Horizontal motion (perpendicular to field): No force acts horizontally, so horizontal velocity \(v_x\) remains constant. If the plates have length \(L\), the time \(t\) spent between the plates is:
\(t = \frac{L}{v_x}\)
• Vertical motion (parallel to field): A constant electric force \(F_E = Eq\) acts along the field lines. According to Newton's Second Law (\(F = ma\)), the vertical acceleration \(a_y\) is:
\(a_y = \frac{F_E}{m} = \frac{Eq}{m}\)
• Vertical displacement (deflection \(y\)): Assuming initial vertical velocity \(u_y = 0\):
\(y = \frac{1}{2} a_y t^2 = \frac{1}{2} \left(\frac{Eq}{m}\right) \left(\frac{L}{v_x}\right)^2\)
Because the vertical displacement depends on \(t^2\) while horizontal distance depends on \(t\), the overall path is a parabola.
Energy and Work Done
Since there is a component of force in the direction of motion, the electric field does work on the particle. This means the kinetic energy changes and the particle speeds up as it deflects.
\(\text{Work Done} = W = F_E \times y = q \Delta V\)
Where \(\Delta V\) is the potential difference through which the particle has moved vertically.
Quick Review: Electric Field Deflection
• Force: \(F_E = Eq\) (constant in magnitude and direction)
• Path: Parabolic
• Speed / Kinetic Energy: Increases (work is done by the field)
2. Charged Particles in a Uniform Magnetic Field
Magnetic fields interact only with moving charges. If a charge sits stationary in a magnetic field, it feels no magnetic force at all!
The Magnetic Force
When a particle carrying charge \(q\) moves at velocity \(v\) perpendicular to a uniform magnetic field of flux density \(B\), it experiences a magnetic force \(F_B\):
\(F_B = Bqv\)
If the particle moves at an angle \(\theta\) to the magnetic field, the general formula is:
\(F_B = Bqv \sin\theta\)
• If the particle moves parallel to the field (\(\theta = 0^\circ\)), \(\sin 0^\circ = 0 \implies F_B = 0\) (no deflection).
• If the particle moves perpendicular to the field (\(\theta = 90^\circ\)), \(\sin 90^\circ = 1 \implies F_B = Bqv\) (maximum deflection).
Determining Direction: Fleming's Left-Hand Rule (FLHR)
To find the direction of the force, hold your left hand with your thumb, first finger, and second finger all at right angles to each other:
• Thumb: Direction of Thrust / Force (\(F\))
• First Finger: Direction of Field (\(B\)) (North to South)
• SeCond Finger: Direction of conventional Current (\(I\))
Crucial Exam Tip: Dealing with Negative Charges
Conventional current is the flow of positive charge.
• For a proton or positive ion: your second finger points in the direction of motion.
• For an electron or negative ion: your second finger must point in the opposite direction to motion!
Trajectory: Circular Motion
According to Fleming's Left-Hand Rule, the magnetic force is always perpendicular to the direction of motion (\(F_B \perp v\)).
Because the force is always at \(90^\circ\) to the velocity:
1. The force can never do any work on the particle (\(W = F d \cos 90^\circ = 0\)).
2. The particle's speed and kinetic energy remain completely constant.
3. The force acts purely as a centripetal force, causing the particle to follow a circular path!
Deriving the Radius of the Circular Path
Setting the magnetic force equal to the centripetal force formula:
\(F_B = F_{\text{centripetal}}\)
\(Bqv = \frac{m v^2}{r}\)
Cancelling one \(v\) from both sides and rearranging for radius \(r\):
\(r = \frac{mv}{Bq} = \frac{p}{Bq}\)
Where \(p = mv\) is the momentum of the particle.
What does this equation tell us?
• Faster or heavier particles (larger momentum \(mv\)) follow a wider circle with a larger radius.
• Stronger magnetic fields (\(B\)) or higher charges (\(q\)) curve the particle more tightly, giving a smaller radius.
Time Period and Frequency of Orbit
The time \(T\) to complete one full circular orbit of circumference \(2\pi r\) is:
\(T = \frac{2\pi r}{v}\)
Substituting \(r = \frac{mv}{Bq}\):
\(T = \frac{2\pi \left(\frac{mv}{Bq}\right)}{v} = \frac{2\pi m}{Bq}\)
Notice something amazing: The speed \(v\) cancels out completely! This means the orbital period \(T\) and frequency \(f = \frac{1}{T} = \frac{Bq}{2\pi m}\) do not depend on the particle's speed or the radius of the orbit. Faster particles simply travel on larger circles in the exact same amount of time. This fundamental principle makes cyclotrons (particle accelerators) work!
Quick Review: Magnetic Field Deflection
• Force: \(F_B = Bqv\) (always perpendicular to velocity)
• Path: Circular (arc of a circle)
• Speed / Kinetic Energy: Constant (zero work is done by the magnetic field)
3. Comparing Electric and Magnetic Deflections
Examiners love to ask you to compare how particles behave in electric versus magnetic fields. Here is a clear summary:
1. Nature of the Force:
• Electric: \(F_E = Eq\). Acts on both stationary and moving charges. Force direction is constant (parallel or antiparallel to field lines).
• Magnetic: \(F_B = Bqv\). Acts only on moving charges. Force direction changes continuously (always perpendicular to velocity).
2. Shape of the Path:
• Electric: Parabolic (like gravity acting on a projectile).
• Magnetic: Circular (acts as a centripetal force).
3. Work Done and Kinetic Energy:
• Electric: Work is done (\(W = q\Delta V\)). Speed and kinetic energy change.
• Magnetic: No work is done (\(W = 0\)). Speed and kinetic energy remain constant.
4. Combined Fields: The Velocity Selector
What happens when we apply both an electric field and a magnetic field in the same region at right angles to each other (crossed fields)?
We can arrange the plates and magnets so that the electric force \(F_E\) and the magnetic force \(F_B\) push in exactly opposite directions on a beam of charged particles.
How the Velocity Selector Works
• The electric field pulls the charge in one direction with force \(F_E = Eq\).
• The magnetic field pushes the charge in the opposite direction with force \(F_B = Bqv\).
For a particle to travel straight through without any deflection, the two forces must balance perfectly:
\(F_E = F_B\)
\(Eq = Bqv\)
Cancelling the charge \(q\) from both sides gives:
\(v = \frac{E}{B}\)
Why is this so useful?
• Only particles with the exact velocity \(v = \frac{E}{B}\) experience balanced forces and pass straight through the exit slit undeflected.
• If a particle moves too fast (\(v > \frac{E}{B}\)): \(F_B > F_E\), so the magnetic force dominates and deflects it into one plate.
• If a particle moves too slow (\(v < \frac{E}{B}\)): \(F_E > F_B\), so the electric force dominates and deflects it into the opposite plate.
• Key Feature: Notice that the selection is independent of the particle's mass and charge! It filters purely based on velocity.
Velocity selectors are used at the front end of mass spectrometers to ensure that all ions enter the deflection chamber with the exact same speed.
5. Common Mistakes to Avoid in Exams
• Forgetting the electron's negative charge in FLHR: Remember, your second finger points opposite to the electron's velocity.
• Confusing trajectory shapes: A uniform electric field produces a parabola; a uniform magnetic field produces a circle (or circular arc).
• Thinking magnetic fields speed up particles: A magnetic force is always perpendicular to velocity, so it changes direction only, never speed.
• Unit consistency: Make sure mass is in kilograms (\(\text{kg}\)), charge in coulombs (\(\text{C}\)), velocity in metres per second (\(\text{m s}^{-1}\)), and magnetic flux density in teslas (\(\text{T}\)).
Summary Checklist
Before moving on, make sure you can confidently:
• Calculate the force on a charge in an electric field (\(F = Eq\)) and describe its parabolic path.
• Calculate the force on a charge in a magnetic field (\(F = Bqv\)) and apply Fleming's Left-Hand Rule.
• Derive the radius of circular motion in a magnetic field (\(r = \frac{mv}{Bq}\)).
• Explain why magnetic fields do no work on charged particles.
• Explain the operation of a velocity selector and derive \(v = \frac{E}{B}\).