In a physics laboratory experiment, a coil with $230$ turns enclosing an area of $14\mathrm{.}0$ cm$2$ is rotated during the time interval $4\mathrm{.}60102$ s from a position in which its plane is perpendicular to Earth$$s magnetic field to one in which its plane is parallel to the field. The magnitude of Earth$$s magnetic field at the lab location is $5\mathrm{.}40105$ T$.$ a$.$ What is the magnitude of the magnetic flux $($initial$)$ through the coil before it is rotated? b$.$ What is the magnitude of the magnetic flux $($final$)$ through the coil after it is rotated? c$.$ What is the magnitude of the average emf induced in the coil?