A rectangular patch of the $xy-$plane is bounded by the lines $x=0,x=L_x,y=L_y,$ and $y=2L_y.E_0$ is a constant with units $\frac{N}{C}.$

$[$Hint: Take dvec$(A)=$dxdyhat$(k).]$

$(50\mathrm{.}1)$ An electric field vec$(E)(t,x,y,z)=E_{0}sin(2t)hat(k)$ fills a region of space enclosing the rectangle. $($i$)$ Compute the rate of change of the electric flux through the rectangle $($ii$)$ Is there necessarily a magnetic field circulating on the boundary of the rectangle?

$(50\mathrm{.}2)$ An electric field vec$(E)(t,x,y,z)=E_{0}sin(2t)hat()$ fills a region of space enclosing the rectangle. $($i$)$ Compute the rate of change of the electric flux through the rectangle $($ii$)$ Is there necessarily a magnetic field circulating on the boundary of the rectangle?

$(50\mathrm{.}3)$ An electric field vec$(E)(t,x,y,z)=\frac{E_0}{\sqrt[\phantom{2}]{L_x{L}_y}}zsin(2t)hat(k)$ fills a region of space enclosing the rectangle. $($i$)$ Compute the rate of change of the electric flux through the rectangle $($ii$)$ Is there necessarily a magnetic field circulating on the boundary of the rectangle?

$(50\mathrm{.}4)$ An electric field vec$(E)(t,x,y,0)=E_0\frac{x}{L_x}(1-\frac{x}{L_x})\frac{y}{L_y}sin(2t)hat(k)$ exists on the rectangular patch. $($i$)$ Compute the rate of change of the electric flux through the rectangle $($ii$)$ Is there necessarily a magnetic field circulating on the boundary of the rectangle?