Saksham Study Point

Magnetic Field Basics

Neutral Point

Where net magnetic field = 0 → Neutral point
Same direction currents → neutral point outside
Opposite currents → neutral point between wires
Field direction by right hand rule

Biot–Savart Law

dB

dB = (μ₀/4π) (I dl sinθ / r²)

dB ⟂ dl and r
Direction: right hand screw rule
μ₀ = permeability of free space

Field due to Straight Wire

Wire

B = μ₀I / (2πr)

Infinite wire case
Semi-infinite: use angle method
Direction circular around wire

Circular Loop

Centre & Axis

Bcenter = μ₀I / (2R)

Baxis = μ₀IR² / 2(R²+x²)^(3/2)

Loop behaves like magnetic dipole
Magnetic moment: M = IA

Ampere’s Law

Loop

∮B·dl = μ₀ I(enclosed)

Used for symmetry cases
Current inside loop only matters

Solenoid

Long

B = μ₀ n I

Inside: uniform field
Outside: ~0
n = turns per unit length

Toroid

Ring

B = μ₀ n I / (2πr)

Inside: field exists
Outside: 0

Magnetic Force

Particle

F = q (v × B)

Perpendicular to velocity
Does no work (changes direction only)

Circular Motion

Charge

r = mv / qB

T = 2πm / qB

Radius ∝ momentum
Time period independent of velocity

Helical Motion

v at angle

Velocity splits into parallel & perpendicular
Pitch = v‖ × T
Radius = mv⊥ / qB

Cyclotron

Accelerator

f = qB / (2πm)

Kmax = q²B²R² / (2m)

Used to accelerate charged particles
Frequency independent of velocity

Force on Conductor

Current

F = I (L × B)

Max when perpendicular
Closed loop in uniform field → net force = 0

Parallel Conductors

Force

F/L = μ₀ I₁I₂ / (2πr)

Same direction → attract
Opposite → repel

Lorentz Force

Total Force

F = q [E + (v × B)]

Includes electric + magnetic force
Velocity selector: v = E/B ⇒ straight path