What Is the Magnetic Pressure & Plasma Beta (β) Calculator?
Plasma Beta (β) is the fundamental dimensionless efficiency parameter of magnetic confinement fusion.
High beta means high fusion power density for a given magnet strength, reducing the cost of electricity.
How Does the Magnetic Pressure & Plasma Beta (β) Calculator Work?
Magnetic fields exert an outward magnetic pressure Pmag = B² / 2μ₀ (~10 MPa at 5 T).
The plasma expands outward with thermal pressure P = 2nkT.
If beta exceeds the Troyon limit (typically 3% to 5% in conventional tokamaks), ballooning and kink MHD modes disrupt the plasma.
Magnetic Pressure & Plasma Beta (β) Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
Compares total thermal fluid pressure of electrons and ions against magnetic energy density.
How to Use the Magnetic Pressure & Plasma Beta (β) Calculator
- Enter core density, temperature in keV, and confining magnetic field.
- Examine the resulting beta percentage against the operational 5% limit.
Step-by-Step Example Calculation
5 T Tokamak at 10 keV and 10²⁰ m⁻³
Input Values:
Understanding Your Result
β ~ 1–5%: Standard operating range for conventional aspect-ratio tokamaks.
β ~ 15–40%: Spherical tokamaks (e.g. MAST-U, NSTX-U) achieving ultra-high beta efficiency.
Factors That Affect the Result
- Magnetic field: Because magnetic pressure scales with B², increasing B decreases beta but allows much higher absolute plasma pressure.
When Should You Use This Calculator?
- Designing magnet coils and structural support jackets for fusion devices.
- Evaluating commercial fusion reactor economic scaling.
Assumptions & Limitations
- Assumes equal electron and ion temperatures (Te = Ti).
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard MHD definition.