📚 Core Concepts
Build from basics to advanced. Master every concept with complete derivations.
1. Lorentz Force & Motion of Charged Particles
1.1 The Fundamental Law
A charged particle moving in electric and magnetic fields experiences the Lorentz Force:
Key Understanding:
- Electric force: F⃗E = qE⃗ (along field direction)
- Magnetic force: F⃗B = q(v⃗ × B⃗) (perpendicular to both v⃗ and B⃗)
The cross product v⃗ × B⃗ ensures the force is always perpendicular to velocity. This means magnetic force can never do work (Work = F⃗·ds⃗ = 0). It only changes direction, not speed.
JEE Twist: Questions often test if you know magnetic force doesn't change kinetic energy.
1.2 Magnitude of Magnetic Force
where θ is the angle between v⃗ and B⃗
Special Cases:
- θ = 0° or 180°: F = 0 (particle moves along field lines)
- θ = 90°: F = qvB (maximum force)
Students forget sin θ and use F = qvB even when velocity isn't perpendicular to field. Always check the angle!
1.3 Direction: Fleming's Left Hand Rule
For positive charge:
- First finger: Magnetic field (B⃗)
- Second finger: Velocity (v⃗)
- Thumb: Force (F⃗)
For negative charge: Force is in opposite direction
1.4 Motion in Uniform Magnetic Field
Case 1: Perpendicular Entry (v⃗ ⊥ B⃗)
Particle moves in a circular path because force is always perpendicular to velocity.
Step 1: Magnetic force provides centripetal force
qvB = mv²/r
Step 2: Solve for r
r = mv/(qB)
Key Insight: Radius ∝ momentum. Heavier or faster particles have larger radius.
Time period is independent of velocity! This is why cyclotron works.
T = 2πr/v = 2π(mv/qB)/v = 2πm/(qB) ✓
Case 2: Oblique Entry (v⃗ makes angle θ with B⃗)
Velocity can be resolved into two components:
- v∥ = v cos θ (parallel to B⃗) → no force, uniform motion
- v⊥ = v sin θ (perpendicular to B⃗) → circular motion
Result: Particle follows a helical path
Questions combine helical motion with energy conservation. Remember: magnetic force doesn't change speed, so kinetic energy remains constant throughout helical path.
1.5 Velocity Selector
Crossed electric and magnetic fields can select particles of specific velocity.
Condition for straight-line motion:
Only particles with this velocity pass undeflected.
2. Biot-Savart Law
2.1 The Fundamental Law
Magnetic field due to a small current element:
where:
- μ₀ = 4π × 10-7 T·m/A (permeability of free space)
- I = current
- dl⃗ = current element (length vector in direction of current)
- r̂ = unit vector from element to point
- r = distance from element to point
| Coulomb's Law | Biot-Savart Law |
|---|---|
| Electric field ∝ charge | Magnetic field ∝ current |
| E ∝ 1/r² | B ∝ 1/r² |
| Field along radial direction | Field perpendicular to current |
2.2 Applications
Direction: Use Right-Hand Thumb Rule
- Thumb points in direction of current
- Curled fingers show direction of magnetic field (circular)
Given current and distance, find field. Or given field and distance, find current. Direct formula substitution. Don't overthink.
where N = number of turns, R = radius
Direction: Perpendicular to plane of coil (right-hand rule)
Students use R² in denominator. It's just R, not R². The r² in Biot-Savart and 1/R cancel out in integration.
where x = distance from center along axis
Special case at center (x = 0):
When x >> R: B ≈ (μ₀NIR²)/(2x³) ≈ (μ₀M)/(2πx³)
This is the dipole field formula! Circular loop acts as magnetic dipole.
where θ is angle subtended in radians
Verification:
- For full circle: θ = 2π → B = (μ₀I)/(2R) ✓
- For semicircle: θ = π → B = (μ₀I)/(4R) ✓
3. Ampere's Circuital Law
3.1 Statement
Line integral of magnetic field around a closed loop equals μ₀ times the current enclosed.
| Use Biot-Savart | Use Ampere's Law |
|---|---|
| Finite wire, arc, any shape | High symmetry (infinite wire, solenoid, toroid) |
| Finding field at specific point | Field at any general point in symmetric case |
| Complex integration needed | Direct answer with symmetry |
3.2 Applications
where n = N/L (turns per unit length)
Key Points:
- Field is uniform inside
- Field is parallel to axis
- Field outside ≈ 0 for ideal solenoid
Derive this using Ampere's law with rectangular loop. Show field outside is zero. 5-mark question guaranteed every year.
where r = distance from center of toroid
Key Points:
- Field exists only inside toroid
- Field inside core = 0
- Field outside toroid = 0
- Toroid = solenoid bent into circle
4. Force on Current-Carrying Conductor
4.1 Force on Straight Wire
where:
- B = magnetic field
- I = current in wire
- L = length of wire in field
- θ = angle between current direction and field
Current I means drift velocity vd of n electrons per unit volume:
I = nAvde
Force on one electron: F1 = evdB
Total electrons in length L: nAL
Total force: F = (nAL) × (evdB) = B(nAvde)L = BIL ✓
4.2 Force Between Parallel Wires
Two parallel wires carrying currents exert force on each other:
Direction:
- Same direction currents: Attractive force
- Opposite direction currents: Repulsive force
One ampere is that current which, when flowing through two infinitely long parallel conductors 1 meter apart in vacuum, produces a force of 2 × 10-7 N/m between them.
This comes directly from F/L = (μ₀I₁I₂)/(2πd)
5. Torque on Current Loop & Magnetic Dipole
5.1 Torque on Rectangular Loop
where:
- N = number of turns
- I = current
- A = area of loop
- B = magnetic field
- θ = angle between normal to loop and field
5.2 Magnetic Dipole Moment
| Electric Dipole | Magnetic Dipole |
|---|---|
| p⃗ = q × d⃗ | M⃗ = I × A⃗ |
| τ⃗ = p⃗ × E⃗ | τ⃗ = M⃗ × B⃗ |
| U = -p⃗·E⃗ | U = -M⃗·B⃗ |
5.3 Potential Energy
Special Cases:
- θ = 0°: U = -MB (minimum, stable equilibrium)
- θ = 90°: U = 0
- θ = 180°: U = +MB (maximum, unstable equilibrium)
6. Applications
Principle:
When current flows through a coil in magnetic field, it experiences a torque and rotates.
where k = torsional constant of spring
Key Features:
- Uses radial magnetic field (B always ⊥ to coil plane)
- Deflection proportional to current
- High sensitivity for small currents
Current Sensitivity:
Voltage Sensitivity:
Q: Explain construction and working of moving coil galvanometer. Derive expression for current sensitivity.
This question appears every year. Practice diagram + derivation.
Conversion to Ammeter:
Connect low resistance (shunt S) in parallel:
Conversion to Voltmeter:
Connect high resistance in series:
Principle:
Charged particle accelerated repeatedly by alternating electric field, moving in circular path due to magnetic field.
Key Concept: Cyclotron frequency is independent of speed
Maximum Energy:
where R = radius of dee
Q: Why doesn't cyclotron work for electrons?
A: At high speeds, relativistic mass increases, so f = qB/(2πm) changes. Particle goes out of phase with alternating voltage.