We can produce current along a wire by immersing it into a magnetic vector field that is varying in time. Faraday's law relates the curl of the electric field E with changing magnetic field B as OB (x, y, Z, t) curlE = at The magnetic field in 3-space is given by B(x, y, z, t) = [-xi-xyj + 3xzk ]sin(t) where x,y,z are the coordinates (according to a rectangular coordinate system) and t is time. The flux of the magnetic field through an immersed surface can be calculated as 2=[[ B.nas The change of the flux of the magnetic field with respect to time is equivalent to the work done by the electric current to circulate though the wire (shown in red colour in the figure below), i.e. Work = - 192 6 E. dr dt At 1 5seconds using Faraday's law, calculate the work done to circulate the electric current through the closed wire, i.e. Work = QE . dr. The wire circulates the part of the cone in the first octant, that is cut-off by the plane y=3x+4 (see figure below). The curve C (shown in red) is circular, and it is the boundary of the surface S (shown in blue) that is the cut-off part of the cone by the plane y=3x+4. The normal of the surface S is oriented in the positive y-direction. That is; the orientation of the surface S is such that the y component of the normal is positive (rather than negative as either of the two can be selected as normal). The cone's tip point is at P(-2,8,6). The vector VR = (1-C, 7-c,, 1 1-cz ) is a position vector between the centre of the circle and the point R(1, 7,11) . Direction of the vector VR is parallel to the z axis and its size is equal to the radius of the circle. The plane that contains the base of the cone Magnetic vector field y=3x+4