(a) Faraday's Law is:
Since the normal to the plane of the coil is parallel to the magnetic field, and . Thus,
From the graph, you can see that the magnetic field as a function of time is linear. The equation of a straight line is ; therefore, since the intercept is 2 T and the slope is , the magnetic field is . You can check this by substituting and calculating which comes out to zero, as expected from the graph.
Applying Faraday's Law gives:
(b) To calculate the current through the coil, apply Ohm's law. The voltage around the coil is equal to the emf.
(c) Because the emf and the current are positive, this means that from the perspective of the normal to the coil, the current flows counterclockwise around the loop. Or consider Lenz's Law, the induced electric field curls around . The magnetic field at the surface of the coil is in the +x direction. It is decreasing (as shown by the graph), thus is opposite and points to the left, in the x direction. As a result, is in the +x direction. Point your thumb in the direction of and your fingers curl around the coil, pointing out of the page at the top of the coil and into the page at the bottom of the coil, as shown below.
Figure: Direction of induced current in the coil.
(d) Point P is in a +z plane. The electric field within the coil curls around the magnetic field in the direction given by the righthand rule. Point your thumb in the direction of and your fingers curl around the coil, pointing out of the page at the top of the coil and into the page at the bottom of the coil, as shown below.
Figure: Direction of the electric field at point P
Note that the curly electric field exists at point P, whether or not their is a coil present. The changing magnetic field induces an electric field regardless of the presence of the coil. Having a coil means that the electric field will exert a force on electrons in the coil, which causes current to flow.
