How to calculate the induced voltage?
The rate of change of flux is directly proportional to the voltage induced in the coil.
Here you say have a solid conductor c connected with terminal hence some current is flowing through it,this current will be opposed by the conductors internal resistance(which will be more in case of the sheet than a wire assuming area of cross-section is fixed),knowing the current in the sheet(derived from the battery) and using Amphere circuital law the magnetic field radiating from the conductor can be calculated.


As you have stated the Emf induced is directly proportional to the product of velocity and the Magnetic field of conductor c will give you the EMF.

The current will flow through wire as load lies in the way it will flow through the load also as it is the only way a closed path can be formed.

Assuming that you are inducing magnetic fields by using the conductor C, the voltage difference across it must be what is across the battery terminals.

Resistance in the magnetic field?
It is called reluctance and it a property of the  conductor which depend upon the physical parameters, you can google to find the formula for various shapes.

The concept usually depends on flux coupling the more flux you can get through a surface the more voltage will be induced, but the flux introduced must be differential with respect to time.

For more information I suggest you can watch lectures of Sir Walter Lewin of MIT 8.01 on electromagnetics.