Magnetic Force: Motors and MRI Machines
3,400 searches/month for "magnetic force" — because this fundamental force powers:
- Electric motors: Fans, hard drives, EV propulsion (F = BIL creates rotation)
- MRI machines: 3 Tesla magnetic field (60,000× Earth's field) images soft tissue
- Speakers: Voice coil moves cone via magnetic force (audio waves)
- Maglev trains: 400 km/h levitation (magnetic repulsion overcomes gravity)
Magnetic force is the interaction between magnetic fields and moving charges (electric current). The key equation:
F = B × I × L × sin(θ)
Where:
- F = Force (Newtons)
- B = Magnetic field strength (Tesla)
- I = Current (Amperes)
- L = Length of conductor in field (meters)
- θ = Angle between current and field (90° = maximum force)
This isn't theoretical — it's the principle behind every motor, generator, speaker, relay, and MRI machine ever built.
The Core Physics: How Magnetic Force Works
The Right-Hand Rule
To find force direction:
- Point fingers in direction of current (I)
- Curl fingers toward direction of magnetic field (B)
- Thumb points in direction of force (F)
Why this matters: Force is perpendicular to both current and field. This creates rotation in motors (force pushes conductor sideways).
The Fundamental Equation
Force on current-carrying conductor in magnetic field:
F = B × I × L × sin(θ)
Maximum force (perpendicular):
F = B × I × L (when θ = 90°)
Zero force (parallel):
F = 0 (when θ = 0°, current parallel to field)
Quick Reference: Magnetic Force in Devices
| Device | B (Tesla) | I (Amps) | L (meters) | F (Newtons) | Application |
|---|---|---|---|---|---|
| Speaker | 1.0 | 0.5 | 0.02 | 0.01 | Voice coil movement |
| DC motor (toy) | 0.5 | 2.0 | 0.05 | 0.05 | Rotation torque |
| EV motor (Tesla) | 1.2 | 400 | 0.3 | 144 | Propulsion force |
| MRI machine | 3.0 | 0.001 | 0.01 | 0.00003 | Hydrogen atom alignment |
| Hard drive | 0.1 | 0.05 | 0.001 | 0.000005 | Read/write head positioning |
| Relay (12V) | 0.3 | 0.1 | 0.01 | 0.0003 | Contact switching |
Key insight: Higher field (B) or higher current (I) = stronger force. Motors use high current; MRI uses ultra-high field.
Real-World Application 1: Electric Motors (How Rotation Happens)
DC Motor Basics
Components:
- Stator: Permanent magnet creating field B
- Rotor: Coil of wire carrying current I
- Commutator: Switches current direction every half rotation
- Brushes: Transfer current to rotating coil
How it works:
Position 1 (0°):
- Current flows clockwise through coil
- Right side of coil: Force upward (F = BIL)
- Left side of coil: Force downward
- Result: Coil rotates counterclockwise
Position 2 (90°):
- Coil perpendicular to field
- Maximum force (sin(90°) = 1)
- Rotation accelerates
Position 3 (180°):
- Commutator switches current direction
- Force reverses to maintain rotation
- Result: Continuous rotation
Calculating Motor Force (Tesla Model 3)
Tesla Model 3 motor specs:
- Magnetic field: 1.2 T
- Peak current: 400 A
- Conductor length (effective): 0.3 m
- Number of coil turns: 100
Force per conductor:
F = B × I × L
F = 1.2 T × 400 A × 0.3 m = 144 N
Total force (100 turns):
F_total = 144 N × 100 = 14,400 N
Torque (rotational force):
τ = F × r (r = rotor radius = 0.1 m)
τ = 14,400 N × 0.1 m = 1,440 N⋅m
Horsepower:
HP = (τ × RPM) / 5,252
HP = (1,440 × 6,000) / 5,252 = 1,645 HP
Why electric motors feel "instant": Full torque at 0 RPM (magnetic force doesn't depend on rotation speed). Gas engines need RPM to build torque.
Efficiency Advantage
Electric motor efficiency:
- Copper wire losses: 5% (I²R heating)
- Magnetic hysteresis: 3%
- Friction/windage: 2%
- Total: 90% efficient
Gas engine efficiency:
- Thermodynamic limit (Carnot): 37% max
- Real-world: 20-25% efficient
- Loss: 75-80% as heat
Why EVs accelerate faster: 90% of energy → motion vs. 25% in gas cars.
Real-World Application 2: MRI Machines (Medical Imaging)
How MRI Uses Magnetic Force
The principle: Hydrogen atoms (in water/fat) act like tiny magnets. Strong magnetic field aligns them, radio waves flip them, and their realignment produces detectable signals.
MRI magnet strength: 1.5 - 3.0 Tesla (clinical), 7.0 T (research)
For comparison:
- Earth's magnetic field: 50 µT (0.00005 T)
- Refrigerator magnet: 0.005 T
- MRI: 3 T (60,000× Earth, 600× fridge magnet)
The Physics of Hydrogen Alignment
Magnetic moment of hydrogen nucleus (proton):
μ = 1.41×10⁻²⁶ J/T
Alignment force in 3 T field:
F = μ × dB/dx (gradient of magnetic field)
Energy difference between aligned and anti-aligned:
ΔE = 2 × μ × B
ΔE = 2 × 1.41×10⁻²⁶ J/T × 3 T = 8.46×10⁻²⁶ J
Corresponding frequency (Larmor frequency):
f = ΔE / h (h = Planck's constant)
f = 8.46×10⁻²⁶ / 6.626×10⁻³⁴ = 127.7 MHz
Why this matters: MRI radio waves transmit at exactly 127.7 MHz (for 3 T field) to flip hydrogen atoms. Different field strengths require different frequencies.
Safety: Projectile Risk
Force on ferromagnetic object (steel wrench):
Steel wrench:
- Mass: 0.5 kg
- Distance from magnet: 1 meter
- Magnetic susceptibility: χ = 1,000
Approximate force:
F ≈ (χ × V × B × dB/dx) / μ₀
Where:
- V = volume (10⁻⁴ m³)
- B = 3 T
- dB/dx = 30 T/m (typical gradient)
- μ₀ = 4π×10⁻⁷ (permeability of free space)
F ≈ (1,000 × 10⁻⁴ × 3 × 30) / (4π×10⁻⁷)
F ≈ 716,000 N = 160,000 lbs
Result: Wrench accelerates to 100+ mph toward magnet (lethal projectile).
Real incidents:
- 2001: Oxygen tank killed 6-year-old (pulled into MRI bore)
- 2018: Janitor's floor buffer flew into scanner ($200k damage)
- 2023: Hospital bed crushed technician (fractured ribs)
Why MRI rooms have strict metal screening: 3 Tesla magnetic field is always on (superconducting magnet, can't be "turned off" quickly).
Real-World Application 3: Speakers (Magnetic Force Creates Sound)
How Speaker Cones Move
Voice coil design:
- Wire coil: 0.02 m diameter, 50 turns
- Permanent magnet: 1.0 T
- Audio current: 0.5 A peak
Force on voice coil:
F = B × I × L
L = circumference × turns = π × 0.02 m × 50 = 3.14 m
F = 1.0 T × 0.5 A × 3.14 m = 1.57 N
Cone acceleration:
a = F / m (cone mass = 0.01 kg)
a = 1.57 N / 0.01 kg = 157 m/s²
Cone displacement (1 kHz tone):
x = a / (2πf)²
x = 157 / (2π × 1,000)² = 4×10⁻⁶ m = 4 µm
Why speakers distort at high volume: Large displacement (>1 mm) moves coil out of uniform magnetic field, reducing force linearity.
Frequency Response
Low frequencies (bass, 50 Hz):
- Large displacement needed: x ∝ 1/f²
- Force must overcome cone inertia
- Requires powerful magnet + high current
High frequencies (treble, 10 kHz):
- Small displacement: 0.01 µm
- Cone mass limits response (can't vibrate fast enough)
- Solution: Separate tweeter (tiny cone, low mass)
Crossover frequency: 2,000 Hz (where woofer hands off to tweeter)
Real-World Application 4: Maglev Trains (Magnetic Levitation)
How Magnetic Repulsion Lifts 500 Tons
Shanghai Maglev:
- Train mass: 500,000 kg
- Operating speed: 430 km/h (267 mph)
- Levitation gap: 10 mm
- Magnetic field: 0.5 T
Force required to levitate:
F_levitation = m × g
F_levitation = 500,000 kg × 9.8 m/s² = 4,900,000 N
Electromagnetic force (simplified):
F = (B² × A) / (2 × μ₀)
Where:
- B = 0.5 T
- A = coil area (10 m² per carriage, 5 carriages = 50 m²)
- μ₀ = 4π×10⁻⁷ T⋅m/A
F = (0.5² × 50) / (2 × 4π×10⁻⁷)
F = 12.5 / (2.51×10⁻⁶)
F = 4,980,000 N ✓ (close to required)
Power consumption (levitation only):
P = F × v (v = levitation gap change rate ≈ 0.1 m/s)
P = 4,900,000 N × 0.1 m/s = 490,000 W = 490 kW
Total power (propulsion + levitation): ~5,000 kW (5 MW) at cruising speed.
Why Maglev Is Faster Than Wheeled Trains
Friction comparison:
Wheeled train (conventional):
- Rolling resistance: F_r = C_r × m × g
- C_r (steel on steel): 0.001
- F_r = 0.001 × 500,000 × 9.8 = 4,900 N
Maglev:
- Rolling resistance: 0 N (no contact)
- Air resistance (dominant at high speed):
- F_air = ½ × ρ × v² × C_d × A
- At 430 km/h: F_air ≈ 150,000 N
Why wheeled trains can't reach 430 km/h: Aerodynamic drag (air resistance) dominates at high speed. Maglev's advantage is stability, not friction reduction at 400+ km/h.
Common Misconceptions About Magnetic Force
Myth 1: "Magnets attract all metals"
The truth: Only ferromagnetic materials (iron, nickel, cobalt, some alloys).
Non-magnetic metals:
- Aluminum (paramagnetic — weak attraction, not noticeable)
- Copper (diamagnetic — weak repulsion)
- Gold, silver, titanium (non-magnetic)
Why this matters: Aluminum screws safe near MRI. Steel screws are lethal projectiles.
Myth 2: "Stronger magnet = more force"
The truth: Force depends on field strength AND current (F = BIL).
Example:
- Weak magnet (0.1 T) + high current (10 A) = 1 N force
- Strong magnet (1.0 T) + low current (0.1 A) = 0.1 N force
Result: Weaker magnet produces 10× more force.
Myth 3: "MRI magnets can be turned off"
The truth: Superconducting MRI magnets are always on (3 T field persists 24/7).
"Quenching" (emergency shutdown):
- Boils off liquid helium (1,700 liters → gas)
- Costs $50,000-100,000 to refill
- Takes 4-6 weeks to ramp back up
- Only done in life-threatening emergencies
Why always-on? Superconducting coils have zero resistance. Once current starts, it flows forever (until helium boils off).
Practical Calculation Example
Problem: Design a relay (electromagnetic switch)
Requirements:
- Switch 120V, 10A load
- 12V coil supply
- Force required to close contacts: 0.5 N
- Air gap: 2 mm
Step 1: Determine magnetic field needed
Simplified relay model:
- Core material: Steel (μ_r = 1,000)
- Coil cross-section area: 1 cm² = 10⁻⁴ m²
Force equation (magnetic attraction):
F = (B² × A) / (2 × μ₀)
Solve for B:
0.5 N = (B² × 10⁻⁴) / (2 × 4π×10⁻⁷)
B² = 0.5 × 2 × 4π×10⁻⁷ / 10⁻⁴
B² = 1.256×10⁻⁶ / 10⁻⁴ = 0.01256
B = 0.112 T
Step 2: Calculate required ampere-turns
Magnetic field in core:
B = μ₀ × μ_r × (N × I) / L
Where:
- L = magnetic path length = 0.05 m
- μ_r = 1,000 (steel)
0.112 = (4π×10⁻⁷ × 1,000 × N × I) / 0.05
N × I = (0.112 × 0.05) / (4π×10⁻⁷ × 1,000)
N × I = 5.6×10⁻³ / 1.256×10⁻³ = 4.46 A⋅turns
Step 3: Choose coil parameters
Option 1: 100 turns, 0.045 A (45 mA) Option 2: 50 turns, 0.089 A (89 mA)
Calculate resistance for 12V supply:
Option 1 (100 turns, 45 mA):
R = V / I = 12V / 0.045A = 267Ω
Wire length: 100 turns × 0.05 m = 5 m
Wire gauge: 30 AWG (0.255 mm dia) → 0.338 Ω/m × 5 m = 1.69Ω
Problem: Need 267Ω, wire provides 1.69Ω. Must add 265Ω resistor (wastes power).
Option 2 (50 turns, 89 mA):
R = 12V / 0.089A = 135Ω
Wire length: 50 × 0.05 = 2.5 m
Wire gauge: 36 AWG → 2.17 Ω/m × 2.5 m = 5.4Ω
Still need resistor: 135 - 5.4 = 129.6Ω (add 130Ω resistor).
Step 4: Verify power dissipation
Coil power:
P_coil = I² × R = 0.089² × 5.4 = 0.043 W
Resistor power:
P_resistor = I² × R = 0.089² × 130 = 1.03 W
Total power: 1.07 W (acceptable for relay application).
How Calculators Make This Easier
Manual magnetic force calculations involve:
- Vector cross products (right-hand rule)
- Field strength conversions (T ↔ Gauss, A⋅m ↔ Oersted)
- Multi-turn coil calculations (N × I ampere-turns)
- Complex geometry (non-uniform fields, fringing effects)
Modern calculators provide:
- Force calculator (input B, I, L → get F)
- Coil design tool (input force required → get N, I, wire gauge)
- Field strength converter (Tesla ↔ Gauss)
- Magnetic circuit analyzer (reluctance, flux, MMF)
Example scenario: You need to design an electromagnet to lift 10 kg.
Calculator inputs:
- Mass to lift: 10 kg
- Air gap: 5 mm
- Core material: Steel (μ_r = 1,000)
- Supply voltage: 24V
Calculator output:
- Required force: 98 N (10 kg × 9.8 m/s²)
- Magnetic field: 0.35 T
- Coil turns: 200
- Current: 1.2 A
- Wire gauge: 22 AWG
- Coil resistance: 6.4Ω
- Required resistor: 13.6Ω (for 24V supply)
- Power consumption: 28.8 W
Manual calculation would require 15+ minutes with electromagnetic field theory. Calculator: instant and accurate.
Professional use: Engineers use magnetic force calculators for:
- Motor design (torque optimization)
- Relay selection (contact force verification)
- Actuator sizing (solenoid stroke force)
- MRI coil design (field uniformity)
These tools aren't shortcuts — they're industry standards. Motor manufacturers use magnetic circuit calculators for optimal winding design.
Summary: Why Magnetic Force Runs Motors and Medical Imaging
The universal principle: F = BIL — magnetic field exerts force on current-carrying conductor. Perpendicular orientation (right-hand rule) creates maximum force.
Key insights:
- Motors: Magnetic force creates rotation (commutator switches current for continuous spin)
- MRI: Ultra-strong field (3 T) aligns hydrogen atoms for imaging
- Speakers: Voice coil force moves cone (current × field = sound)
- Maglev: Magnetic repulsion levitates train (zero friction)
- Safety: Ferromagnetic objects become lethal projectiles in MRI rooms
Real-world mastery:
- EV motors use 1.2 T field + 400 A current = 144 N force per conductor
- MRI 3 T field is 60,000× Earth's magnetic field (dangerous projectile risk)
- Speaker voice coils move 4 µm at 1 kHz (magnetic force drives all sound)
- Maglev trains levitate 10 mm gap with 0.5 T field (4.9 million Newtons force)
The bottom line: Magnetic force isn't optional in modern technology — it's the fundamental principle behind electric motors, medical imaging, audio reproduction, and high-speed transportation.
Whether you're designing a motor, understanding MRI safety, troubleshooting a speaker, or analyzing maglev physics, F = BIL is the equation that governs magnetic interactions.
This isn't abstract physics — it's the reason Tesla Model 3 accelerates instantly (full torque at 0 RPM), the reason MRI rooms have strict metal policies (3 T field never turns off), and the reason speakers reproduce 20 Hz to 20 kHz (magnetic force moves cone precisely).
Understanding magnetic force transforms everyday devices from "magic" to physics-based systems you can analyze, design, and optimize.