Q:

How to calculate eddy current with NDSolve

I want to calculate the eddy currents inside a plane plate b,h, which moves with constant velocity v nearby a fixed magnet(radius r, magnetic flux normal to the plate) 
My attempt
werte = {b = 1 , h = 1 , r = 0.25, v = 1, T = 0.01};
tsim = b/2/v;
NDSolveValue[{Laplacian[\[Phi][t, x, y], {x, y}] == 
Derivative[1 , 0, 0 ][\[Phi]][t, x, y]
,
\[Phi][0, x, y] == 0 (* initial condition *)
,
DirichletCondition[0, True]  (* if necessary? *)
,
(* magnetic flux normal to the magnet *)
\[Phi][t, x, y] ==If[(v t + x - b/2)^2 + y^2 <=r^2, 1 - Exp[-t/T], 0]}, \[Phi]  
, 
{x,-b/2,b/2}, {y, -h/2, h/2}, {t, 0, tsim},
Method -> {"MethodOfLines", "TemporalVariable" -> t, "SpatialDiscretization" -> {"FiniteElement"}}
]

results in this error message
NDSolveValue::overdet: There are fewer dependent variables, {\[Phi][t,x,y]}, than equations, so the system is overdetermined.

Is there any experinece solving such physical problems? 

A:

You have two equations that are defined for all values of t, x, and y, rather than just on the boundaries:
Laplacian[\[Phi][t, x, y], {x, y}] == Derivative[1 , 0, 0 ][\[Phi]][t, x, y]

\[Phi][t, x, y] == If[(v t + x - b/2)^2 + y^2 <=r^2, 1 - Exp[-t/T], 0] 

Since you have two equations defined on the interior, and only one unknown, the system is overdetermined, which is why Mathematica is giving you this error message.
(Note also that the function $\Phi$ defined in the second equation does not satisfy the PDE given in the first equation, since inside the moving circular region $\nabla^2 \Phi = 0$ but $\dot{\Phi} \neq 0$.)

