Engineering & Science
Alfvén Wave Speed (vA) & Plasma Beta (MHD Magnetic Confinement Formula)
Calculate magnetohydrodynamic Alfvén speed vA = B / SQRT(mu0 * rho) in km/s and dimensionless Plasma Beta = P_thermal / (B^2 / (2 * mu0)) for tokamak fusion reactors and solar-wind plasmas.
Treat magnetic field lines like plucked violin strings loaded with plasma ions!
Hannes Alfvén (1970 Physics Nobel Prize) showed that in an electrically conducting plasma, magnetic field lines carry tension B^2 / mu0 while frozen-in plasma mass density rho (kg/m^3) provides inertia—creating transverse magnetohydrodynamic (MHD) Alfvén waves propagating at speed vA = B / SQRT(mu0 * rho), where mu0 = 4 * PI() * 10^-7 H/m is vacuum permeability!
Simultaneously, tokamak and stellarator fusion physicists track dimensionless Plasma Beta—the ratio of plasma kinetic pressure P_plasma = n * kB * T (Pascals) to magnetic confining pressure P_mag = B^2 / (2 * mu0): beta = P_plasma / (B^2 / (2 * mu0))! When beta << 1 (e.g., solar corona or a 2.0 T tokamak at P_plasma = 40,000 Pa and rho = 1.0e-6 kg/m^3, giving vA = 1,784.12 km/s and beta = 0.02513 = 2.51%), magnetic fields dominate plasma motion; when beta > 1 (solar photosphere / interstellar medium), gas pressure drags magnetic field lines around!
| MHD Plasma State (rho = 10^-6 kg/m^3, P = 40 kPa) | Magnetic Field B (Tesla) | Alfvén Speed vA (km/s) | Plasma Beta |
|---|---|---|---|
| Baseline Tokamak Edge / Discharge | B = 2.0 T | 1,784.12 km/s | beta = 0.02513 (2.51%, Magnetically Confined) |
| High-Field HTS Magnet Upgrade | B = 4.0 T (2x Field) | 3,568.25 km/s (2.0x vA) | beta = 0.00628 (4x higher P_mag!) |
| Invalid Density rho <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid MHD Inputs" |
Alfvén Wave Velocity vA = (B / SQRT(mu0 * rho)) / 1000 (km/s)
Uses exact vacuum permeability mu0 = 4 * PI() * 10^-7 N/A^2 and converts m/s to km/s (/ 1,000).
=IF(OR(B2 < 0, B3 <= 0), "Invalid MHD Inputs", ROUND((B2 / SQRT((4 * PI() * 0.0000001) * B3)) / 1000, 2))Dimensionless MHD Plasma Beta = P_thermal / (B^2 / (2 * mu0))
Divides kinetic plasma pressure B4 (Pa) by magnetic pressure B^2 / (2 * mu0).
=IF(OR(B2 <= 0, B4 < 0), "Invalid MHD Inputs", ROUND(B4 / ((B2 ^ 2) / (2 * 4 * PI() * 0.0000001)), 5))Try the example
Use the original values or change them to test your understanding.