Force on a Charged Particle: Lorentz force equation and motion in combined electric and magnetic fields (velocity selector)

📌 1. Question Description

Force on a Charged Particle

A charged particle moving in electric and magnetic fields experiences electric and magnetic forces. The total force acting on the particle is called the Lorentz force.

F⃗ = q(E⃗ + v⃗ × B⃗)

For a charged particle:

• Electric force: F_E = qE
• Magnetic force: F_B = q(v × B)
• Total Lorentz force: F = q(E + v × B)

Important: The magnetic force is always perpendicular to both the velocity of the particle and the magnetic field.

⚡ 2. Lorentz Force Equation

F⃗ = q(E⃗ + v⃗ × B⃗)

Where:

• q = charge of particle
• E = electric field
• v = velocity of particle
• B = magnetic field

📐 3. Electric Force

When only an electric field is present, the force on a charge is:

F_E = qE

For a positive charge, the electric force is in the direction of electric field. For a negative charge, it is opposite to the electric field.

🧲 4. Magnetic Force

F_B = qvB sinθ

Here θ is the angle between velocity and magnetic field.

θ = 0°

F = 0

Particle moves parallel to magnetic field.

θ = 90°

F = qvB

Magnetic force is maximum.

θ = 180°

F = 0

Particle moves antiparallel to magnetic field.

🔬 5. Animated Practical — Velocity Selector

Charge Positive (+q) Negative (-q)

Velocity v 5.0

Electric Field E 5.0

Magnetic Field B 5.0

VELOCITY SELECTOR

E → ELECTRIC FIELD

B ⊗ MAGNETIC FIELD

Velocity: 5.00
Electric Field: 5.00
Magnetic Field: 5.00
Required velocity: 1.00
Condition: v = E/B
✔ PARTICLE PASSES UNDEFLECTED

🎯 6. Principle of Velocity Selector

A velocity selector uses mutually perpendicular electric and magnetic fields to allow only particles having a particular velocity to pass through undeflected.

For a positively charged particle, electric and magnetic forces act in opposite directions when the particle moves perpendicular to both fields.

F_E = qE

F_B = qvB

For an undeflected particle:

F_E = F_B

Therefore:

qE = qvB

Hence:

⭐ v = E/B

Velocity selected by the selector:

v = E/B

🚀 7. Motion in Electric and Magnetic Fields

Only Electric Field

F = qE

The charged particle accelerates along or opposite the electric field.

Only Magnetic Field

F = qvB sinθ

For v ⟂ B, the particle moves in a circular path.

Both E and B

F = q(E + v × B)

The motion depends upon the relative directions and magnitudes of E, B and v.

⭕ 8. Charged Particle Moving Perpendicular to Magnetic Field

If a charged particle enters a uniform magnetic field perpendicular to the field, magnetic force provides the centripetal force.

qvB = mv²/r

Therefore:

⭐ r = mv/qB

Angular frequency:

ω = qB/m

Time period:

⭐ T = 2πm/qB

🌀 9. Helical Motion

If the particle enters the magnetic field at an angle other than 0° or 90°, its velocity can be resolved into two components:

v_∥ = v cosθ

v_⊥ = v sinθ

The perpendicular component produces circular motion while the parallel component produces uniform motion along the magnetic field. Hence the particle follows a helical path.

🧭 10. Direction of Magnetic Force

The direction of magnetic force is determined using the right-hand rule for a positive charge.

For a negative charge, the direction is opposite to that obtained from the right-hand rule.

F⃗_B = q(v⃗ × B⃗)

📊 11. Velocity Selector Conditions

Condition	Result
v = E/B	Particle passes undeflected
v > E/B	Magnetic force dominates
v < E/B	Electric force dominates
E = 0	Only magnetic force acts
B = 0	Only electric force acts

📝 12. MCQ Practice

1. The Lorentz force on a charged particle is:

A. qE

B. qvB

C. q(E + v × B)

D. q/B

✔ Answer: C

2. Magnetic force on a charged particle is maximum when the angle between v and B is:

A. 0°

B. 30°

C. 60°

D. 90°

✔ Answer: D

3. Magnetic force on a stationary charged particle is:

A. qB

B. qE

C. Zero

D. Infinite

✔ Answer: C

4. A velocity selector selects particles according to their:

A. Charge

B. Mass

C. Velocity

D. Energy only

✔ Answer: C

5. The velocity of an undeflected particle in a velocity selector is:

A. EB

B. E/B

C. B/E

D. E+B

✔ Answer: B

6. Magnetic force does no work because it is:

A. Parallel to velocity

B. Perpendicular to velocity

C. Parallel to magnetic field

D. Zero always

✔ Answer: B

7. A charged particle entering a magnetic field perpendicular to it follows:

A. Straight path

B. Circular path

C. Parabolic path

D. Elliptical path

✔ Answer: B

8. Radius of circular path of a charged particle is:

A. mv/qB

B. qB/mv

C. qv/mB

D. mB/qv

✔ Answer: A

9. If electric and magnetic forces on a particle are equal and opposite, then:

A. Particle accelerates

B. Particle is undeflected

C. Particle stops

D. Particle gains charge

✔ Answer: B

10. The magnetic force depends on:

A. q, v and B

B. q only

C. B only

D. mass only

✔ Answer: A

🟣 13. Assertion–Reason — 5 Questions

Options:
A. Both A and R are true, and R is the correct explanation of A.
B. Both A and R are true, but R is not the correct explanation of A.
C. A is true, but R is false.
D. A is false, but R is true.

Assertion: A magnetic field cannot change the kinetic energy of a charged particle.

Reason: Magnetic force is perpendicular to the instantaneous velocity of the particle.

Answer: A

Assertion: A charged particle entering a uniform magnetic field perpendicular to it moves in a circular path.

Reason: Magnetic force acts as the centripetal force.

Answer: A

Assertion: A velocity selector allows particles with only one particular velocity to pass undeflected.

Reason: For an undeflected particle, qE = qvB.

Answer: A

Assertion: The magnetic force on a stationary charged particle is zero.

Reason: Magnetic force is proportional to velocity.

Answer: A

Assertion: A negative charge experiences magnetic force in the same direction as a positive charge moving with the same velocity.

Reason: Magnetic force depends on the sign of charge.

Answer: D

🟢 14. 2 Marks — 6 Questions

Q1 Write the Lorentz force equation.

Q2 Write the expression for magnetic force on a charged particle moving in a magnetic field.

Q3 When is the magnetic force on a charged particle maximum?

Q4 What is the principle of a velocity selector?

Q5 Write the velocity selected by a velocity selector.

Q6 Why does magnetic force do no work on a charged particle?

🟡 15. 3 Marks — 6 Questions

Q1 Explain the Lorentz force acting on a charged particle in combined electric and magnetic fields.

Q2 Derive the condition for a charged particle to pass undeflected through a velocity selector.

Q3 Explain the motion of a charged particle entering a uniform magnetic field perpendicular to it.

Q4 Derive the radius of circular motion of a charged particle in a magnetic field.

Q5 Explain why a magnetic field changes the direction but not the speed of a charged particle.

Q6 Explain the role of electric and magnetic fields in a velocity selector.

🟠 16. 4 Marks — 6 Questions

Q1 State and explain the Lorentz force equation.

Q2 Derive the expression v = E/B for a velocity selector.

Q3 Derive the radius of circular path followed by a charged particle in a uniform magnetic field.

Q4 Explain the different types of motion of a charged particle in a magnetic field for θ = 0°, 90° and intermediate angles.

Q5 A particle passes undeflected through electric field E = 2 × 10⁴ N/C and magnetic field B = 0.5 T. Find its velocity.

Q6 Explain why a velocity selector is useful for selecting charged particles of a definite velocity.

🔴 17. 5 Marks — 6 Questions

Q1 Explain Lorentz force and derive its expression for a charged particle moving in combined electric and magnetic fields.

Q2 Explain the working principle of a velocity selector and derive v = E/B.

Q3 Derive the expression for radius and time period of circular motion of a charged particle in a uniform magnetic field.

Q4 Explain the motion of a charged particle in a magnetic field when it enters at an arbitrary angle.

Q5 A proton enters a magnetic field of 0.2 T with speed 4 × 10⁶ m/s perpendicular to the field. Derive the expression and calculate the radius of its path.

Q6 Explain why the magnetic field does no work while the electric field can change the kinetic energy of a charged particle.

🔵 18. 6 Marks — 6 Questions

Q1 State and derive the Lorentz force equation. Explain separately the electric and magnetic components of the force.

Q2 Explain the construction, principle and working of a velocity selector. Derive the expression v = E/B with a suitable diagram.

Q3 Derive the radius, angular frequency and time period of a charged particle moving perpendicular to a uniform magnetic field.

Q4 Explain the complete motion of a charged particle in a uniform magnetic field when its velocity is parallel, perpendicular and inclined to the field.

Q5 A charged particle enters mutually perpendicular electric and magnetic fields. Derive the condition under which it passes undeflected and explain how the arrangement acts as a velocity selector.

Q6 Explain Lorentz force, velocity selector, circular motion and helical motion of charged particles with necessary equations.

🧮 19. Numerical Practice

Example 1 — Velocity Selector

Electric field: E = 2 × 10⁴ N/C

Magnetic field: B = 0.5 T

v = E/B

v = (2 × 10⁴)/0.5

⭐ v = 4 × 10⁴ m/s

✔ Answer: 4 × 10⁴ m/s

Example 2 — Circular Motion

A particle of mass m and charge q enters a magnetic field B with velocity v perpendicular to the field.

qvB = mv²/r

⭐ r = mv/qB

Thus the radius increases with mass and velocity and decreases with charge and magnetic field.

📚 20. Important Formula Sheet

Lorentz Force:
F⃗ = q(E⃗ + v⃗ × B⃗)

Electric Force:
F_E = qE

Magnetic Force:
F_B = qvB sinθ

Velocity Selector:
v = E/B

Circular Radius:
r = mv/qB

Angular Frequency:
ω = qB/m

Time Period:
T = 2πm/qB

🚀 21. Quick Revision

⚡ Lorentz Force

F = q(E + v × B)

🧲 Magnetic Force

F = qvB sinθ

🎯 Velocity Selector

v = E/B

⭕ Circular Motion

r = mv/qB

🌀 Helical Motion

Occurs when v has both parallel and perpendicular components.

💡 Key Point

Magnetic force does no work on a charged particle.