Part II$.$ Relationship between electric and magnetic fields $($Faraday$\text{'}$s law$)$

Background

One fascinating scientific discovery of the ${19}^{th}$ century was the realization that a varying magnetic field in time can produce electric current. This physical phenomenon was discovered by Michael Faraday, the British physicist and chemist $($pictured$),$ and made the foundation for the invention of electric motors. He also realized that the change of the flux of a magnetic field with respect to time is equivalent to the work done by the electric current to circulate through a wire that is immersed inside the magnetic field. Each and every electric motor "works" based on these two fundamental principles.

According to Faraday's law, an unsteady magnetic field B $($a magnetic field that changes in time$)$ is associated with an electric field $E$ based on the following equation

grad$E=\frac{\text{d}\text{e}\text{l}\text{B}(x,y,z,t)}{\text{d}\text{e}\text{l}\text{t}}$

In addition, the flux of the magnetic field $B$ through an immersed surface can be calculated as:

$={}_{S}B*ndS$

In which $$ is the flux of the magnetic field $B$ through an immersed surface $S$ with the normal vector $n.$ Accordingly, the work done by the electric current to circulate though the wire can be quantified as:

work $=-\frac{d}{dt}=o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}E*dr$

Problem

A $3-$dimensional magnetic field is given as:

$B(x,y,z,t)=[-x^{2}i-xyj+3xzk]sint$

where $x,y,z$ are the space coordinates and $t$ is time. Using Faraday's law, calculate the work done by B at $t=5$ to circulate the electric current through the closed red wire $C,$ shown in Figure $1$ on the next page. The wire $C$ is part of the cone in the first octant, that is cut$-$off by the plane $S$:$y=3x+4($see figure$).$ The normal of the surface $S$ is oriented in the positive $y-$direction. The cone's tip is at $P(-2,8,6).$ The vector $V_R=(1-c_x,7-c_y,11-c_z)$ is a position vector between the centre of the

circle and the point $R(1,7,11)$ over the wire. Direction of the vector $V_R$ is parallel to the $z$ axis and the size of the vector $V_R$ is equal to the radius of the circle.

Figure $1.$ The schematic illustrating the electric motor introduced in the problem for Part II$.$