Ghostly electric field

The space within the cylinder contains a time varying magnetic field distributed uniformly. Find the work done by an external force to carry a charge ${ q }_{ 0 }$ from the point $A$ along the shown path to infinity, quasi statically. Give the value of $k$ in ${ { ka }_{ 0 } }^{ 2 }{ { B } }_{ 0 }{ q }_{ 0 }$ is a real number.