Theory: Eddy Current Brake Simulator
When a conductor moves through a magnetic field, the magnetic flux through microscopic current loops changes.
By Faraday’s law, this produces an induced electric field (an EMF), and because the material has finite conductivity
\( \sigma \), it drives circulating currents called eddy currents.
By Lenz’s law, the induced currents create a magnetic field that opposes the change that created them—resulting in a
resistive force (or torque) that opposes motion.
Key idea
Eddy-current braking often behaves like a speed-proportional drag at moderate speeds:
\[
F_{\text{brake}} \approx c\,v
\quad\text{(translation)},
\qquad
\tau_{\text{brake}} \approx c_\omega\,\omega
\quad\text{(rotation)}.
\]
The effective coefficients \(c\) and \(c_\omega\) depend strongly on geometry, thickness, conductivity, field strength,
and how “closed” the current loops can be. In real systems, at high frequencies/speeds, skin effect changes the scaling.
Simple linear-drag model used here
This simulator uses a qualitative “linear drag” approximation with an adjustable empirical factor \(C_0\),
to connect the inputs \( \sigma \) and \(B\) to a plausible drag coefficient:
\[
c \;=\; C_0\,\sigma\,B^2\,t\,A^{3/2}
\quad\Rightarrow\quad
m\,\frac{dv}{dt}=-c\,v
\]
\[
v(t)=v_0\,e^{-t/\tau},\quad \tau=\frac{m}{c}.
\]
For the rotating disk option, the same idea is applied to a speed-proportional torque:
\[
c_\omega \;=\; C_0\,\sigma\,B^2\,t\,R^4
\quad\Rightarrow\quad
I\,\frac{d\omega}{dt}=-c_\omega\,\omega
\]
\[
\omega(t)=\omega_0\,e^{-t/\tau_\omega},\quad \tau_\omega=\frac{I}{c_\omega}.
\]
Power dissipation
The braking converts mechanical energy into heat. With linear drag:
\[
P(t)=Fv=c\,v^2
\quad\text{or}\quad
P(t)=\tau\omega=c_\omega\,\omega^2.
\]
For pure linear drag, the dissipated energy over time matches the kinetic energy drop:
\( \Delta E = \tfrac12 m(v_0^2-v^2) \) (or rotational analog \( \tfrac12 I(\omega_0^2-\omega^2) \)).
Important note
Real eddy-current brakes depend on exact geometry, field region size, gaps, and frequency effects.
This tool is designed to build intuition (direction + scaling) and provide clean time-constant behavior.
