Moving Charges & Magnetism: Biot–Savart, Ampere & Lorentz Force
Study how moving charges produce magnetic fields, learn Biot–Savart law, Ampere's circuital law, Lorentz force, cyclotron, solenoid and toroid for JEE Main Physics.
The central idea of this chapter
Moving charges create magnetic fields, and magnetic fields exert forces on other moving charges. This chapter connects current, field and force through Biot–Savart, Ampere and Lorentz force.
What should you understand first?
Magnetism due to currents is the core of this chapter. Once you understand field patterns and right-hand rules, most problems reduce to applying standard results for wires, loops, solenoids and toroids.
Core concepts
- Magnetic field & field lines B-vector idea
- Biot–Savart law dB from current element
- Ampere's circuital law ∮B·dl = μ₀I
- Lorentz force F = q(v × B)
- Cyclotron motion Circular path in B
Useful building blocks
- Straight wire B = μ₀I/(2πr)
- Circular loop On axis & centre
- Solenoid B = μ₀nI
- Toroid B = μ₀NI/(2πr)
- Force on conductor F = I(l × B)
Moving Charges & Magnetism formula sheet
Use right-hand thumb rule and screw rule consistently. Memorise standard field results and apply Ampere's law only where symmetry allows.
| Topic | Formula / Relation | Meaning or use |
|---|---|---|
| Biot–Savart law | dB = (μ₀/4π) · (I dl sinθ / r²) | Magnetic field due to small current element |
| Field due to straight wire | B = μ₀I/(2πr) | Long straight conductor at perpendicular distance r |
| Field at centre of loop | B = μ₀I/(2R) | Circular loop of radius R carrying current I |
| Field on axis of loop | B = (μ₀IR²)/(2(R² + x²)^(3/2)) | At distance x from centre on the axis |
| Ampere's circuital law | ∮B·dl = μ₀Ienclosed | Line integral of B around closed loop |
| Solenoid (inside) | B = μ₀nI | n is number of turns per unit length |
| Toroid (inside) | B = μ₀NI/(2πr) | N total turns, r is mean radius |
| Lorentz force (charge) | F = q(v × B) | Force on charge q moving in magnetic field |
| Lorentz force (conductor) | F = I(l × B) | Force on current-carrying conductor of length l |
| Cyclotron frequency | f = qB/(2πm) | Frequency of circular motion in uniform B |
| Radius of circular path | r = mv/(qB) | For charge moving perpendicular to B |
| Force between parallel wires | F/l = μ₀I₁I₂/(2πd) | Force per unit length between two parallel currents |
How to approach magnetism problems
Identify the geometry first (wire, loop, solenoid, toroid), then use the appropriate standard result or Ampere's law. Apply Lorentz force for motion and force questions.
Field calculations
- Straight wire? B = μ₀I/(2πr)
- Circular loop? Use axis formula
- High symmetry? Apply Ampere's law
- Solenoid or toroid? Use standard B
- Multiple conductors? Vector sum of B
Force & motion
- Charge in B-field? F = q(v × B)
- Conductor in B-field? F = I(l × B)
- Circular motion? r = mv/(qB)
- Cyclotron? f = qB/(2πm)
- Parallel wires? F/l formula
How to prepare Moving Charges & Magnetism
Start with Biot–Savart and field patterns, then learn Ampere's law applications. Finish with Lorentz force, cyclotron and force between conductors.
What to do
- Learn Biot–Savart law and practise field due to wire and loop
- Master Ampere's law for solenoid, toroid and symmetric cases
- Solve Lorentz force problems for charges and current-carrying wires
- Understand cyclotron principle, frequency and radius
- Revise force between parallel conductors and torque on loop
Common mistakes
- Using wrong direction in right-hand thumb rule or cross product
- Applying Ampere's law where symmetry is not sufficient
- Confusing field at centre of loop with field on its axis
- Forgetting that B inside an ideal solenoid is uniform and axial
- Mixing up formulas for solenoid and toroid
Ready to test Moving Charges & Magnetism?
Revise the formula sheet, then solve mixed JEE Main problems on magnetic field, Lorentz force and instruments.
Moving Charges & Magnetism FAQ
Short answers to frequently tested ideas in this chapter.
The Biot–Savart law gives the magnetic field due to a small current element. It states that dB is proportional to I dl sinθ and inversely proportional to r², directed perpendicular to the plane of dl and r.
Ampere's circuital law states that the line integral of magnetic field around a closed loop equals μ₀ times the net current enclosed by that loop. It is written as ∮B·dl = μ₀I_enclosed.
The Lorentz force on a charge q moving with velocity v in a magnetic field B is F = q(v × B). The force is perpendicular to both velocity and magnetic field.