Proton-like ion at 60 degrees
A 1 u, +1 e particle at 500 km/s in 200 mT has 75% of its kinetic energy perpendicular to B and 25% parallel, producing finite helical radius.
Physics and electromagnetism
Magnetic Force Energy Calculator
Partition a nonrelativistic charged particle’s kinetic energy into parallel and perpendicular components and calculate its Larmor radius in a uniform magnetic field.
Uniform-field energy partition
Separate magnetic turning from actual energy transfer
A static magnetic field bends the velocity component perpendicular to B but does no work. This page quantifies total, perpendicular, and parallel kinetic energy and connects the perpendicular share to the Larmor radius.
Total kinetic energy-
Perpendicular energy-
Parallel energy-
Larmor radius-
Current model evidence
Kinetic-energy and gyroradius ledger
Use the energy closure to audit the pitch-angle split and the radius only for the perpendicular motion.
| Energy stage | Primary basis | Secondary basis | Current value | Scope / unit |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
K = mv^2/2; K_perp = K sin^2(theta); K_parallel = K cos^2(theta); rL = m v_perp/(|q|B)
Convert mass, charge, speed, and field to SI; resolve velocity by pitch angle; compute kinetic-energy shares; derive the Larmor radius from perpendicular momentum; verify that the shares close to total energy.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| m | Particle rest mass | kg | 1 u |
| q | Signed particle charge | C | +1 e |
| v | Particle speed | m/s | 500 km/s |
| theta | Pitch angle from B | deg | 60 deg |
| K_perp | Kinetic energy in perpendicular motion | J | Solved |
| rL | Larmor radius | m | Solved |
3. Unit and sign normalization
- Mass in u is multiplied by the CODATA atomic mass constant.
- Charge number is multiplied by the exact elementary charge; radius uses its absolute value.
- Speed is multiplied by 1000 and B by 1e-3 before energy and radius calculations.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Five steps from measured inputs to a defensible result
- Select a particle mass in atomic mass units and preserve the actual ionization state.
- Enter the signed charge number; the magnitude controls radius while the sign controls rotation direction outside this page.
- Enter speed below 0.1c and a nonzero uniform B magnitude.
- Measure pitch angle from the field direction, not from a plane perpendicular to B.
- Use the closure residual and stated limits before transferring the radius to aperture or confinement decisions.
ELECTROMAGNETIC FOUNDATIONS
Concepts that control this specific model
- Magnetic work is zero
- For F = q(v x B), F is perpendicular to v, so instantaneous power F dot v vanishes.
- Pitch angle partitions velocity
- v_perp = v sin(theta) gyrates; v_parallel = v cos(theta) advances along the field.
- Energy shares use squared components
- The sin^2 and cos^2 terms ensure K_perp + K_parallel = K.
- Radius follows perpendicular momentum
- Only m v_perp enters rL because parallel motion does not bend around the field line.
- Charge sign does not change radius magnitude
- Replacing q by -q reverses the rotation sense but leaves |q| and rL unchanged.
DEEP ANALYSIS 1
A helical path can have constant energy
Direction changes continuously even when speed and kinetic energy do not. Force and energy transfer should therefore be reported separately.
DEEP ANALYSIS 2
Pitch-angle endpoints are meaningful boundaries
At 0 or 180 degrees, v_perp and rL are zero; at 90 degrees all kinetic energy is perpendicular and the guiding center does not advance along B.
DEEP ANALYSIS 3
The 0.1c limit protects the classical approximation
At higher speeds relativistic momentum increases the radius and changes frequency. Rejecting those values is safer than silently applying mv.
RESULT INTERPRETATION
What the current output does and does not decide
Total energy is the classical translational kinetic energy before and during ideal uniform-field motion, not energy stored in the magnet.
Larmor radius is a local uniform-field scale; it is not a full orbit radius when electric fields, gradients, collisions, or changing B are important.
REAL USE CASES
Two decisions with different boundary conditions
Field-aligned injection
At pitch angle 0 degrees the particle keeps all kinetic energy in parallel motion and the ideal Larmor radius is zero because magnetic force vanishes.
EVIDENCE AND DATA QUALITY
What to retain with the exported result
Retain particle species and charge state, speed-estimation method, pitch-angle definition, local field calibration, time and location, and the exported energy closure. State explicitly whether relativistic or electric-field effects were screened out.
LIMITS AND EXCLUSIONS
Where this physical model stops
- Uses nonrelativistic kinetic energy and rejects speed at or above 0.1c.
- Assumes a uniform static magnetic field and no electric field.
- Excludes radiation, collisions, magnetic moments, field gradients, and adiabatic invariants.
- Treats mass and charge state as fixed and exactly known.
- Does not calculate orbit phase, guiding-center drift, or confinement lifetime.
TERMS USED HERE
Six terms that keep the calculation unambiguous
- Pitch angle
- Angle between particle velocity and magnetic-field direction.
- Perpendicular energy
- Kinetic energy associated with velocity transverse to B.
- Parallel energy
- Kinetic energy associated with velocity along B.
- Larmor radius
- Radius of ideal circular gyromotion about a field line.
- Guiding center
- Center around which the perpendicular velocity gyrates.
- Magnetic work
- Energy-transfer rate F dot v, equal to zero for the ideal magnetic force.
RELIABLE SOURCES
References supporting the equation and units
FREQUENTLY ASKED QUESTIONS
Questions specific to this calculation
Why does the field not appear in total kinetic energy?
A static magnetic field changes velocity direction, not speed, so K depends on mass and speed only.
Why does B affect the radius?
A stronger field supplies greater transverse force at the same speed, bending the path more tightly.
What happens at a 90-degree pitch angle?
All motion is perpendicular, K_perp equals K, K_parallel is zero, and the ideal path is circular.
Does negative charge produce negative energy?
No. Energy and radius are nonnegative; charge sign only reverses gyromotion direction.
Why is zero charge rejected?
An uncharged particle has no magnetic gyromotion, making the Larmor-radius denominator and this model inapplicable.
Can I use an electron mass in u?
Yes, if entered accurately, but the same nonrelativistic speed limit and fixed-mass assumptions apply.
IMPORTANT BOUNDARY
Use the result as analysis, not certification
This ideal single-particle model is not a plasma-confinement design, radiation analysis, accelerator acceptance study, or relativistic orbit calculation.