To work through a straightforward application of Faraday's law to find the EMF and the electric field surrounding a region of increasing fluxFaraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. This problem is a prototypical example in which an increasing magnetic flux generates a finite line integral of the electric field around a closed loop that surrounds the changing magnetic flux through a surface bounded by that loop. A cylindrical iron rod with cross-sectional area is oriented with its symmetry axis coincident with the z axis of a cylindrical coordinate system as shown. It has a uniform magnetic field inside that varies according to . In other words, the magentic field is always in the positive z direction, and it has no other components. For your convenience, we restate Faraday's law here: , where is the line integral of the electric field, and the magnetic flux is given by , where is the angle between the magnetic field and the local normal to the surface bounded by the closed loop. Direction: The line integral and surface integral reverse their signs if the reference direction of or is reversed. The right-hand rule applies here: If the thumb of your right hand is taken along , then the fingers point along . You are free to take the loop anywhere you choose, although usually it makes sense to choose it to lie along the path of the circuit you are considering. Find , the electromotive force (EMF) around a loop that is at distance from the z axis, where is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as positive. Express in terms of , , , , and any needed constants such as , , and .Due to the cylindrical symmetry of this problem, the induced electric field can depend only on the distance from the z axis, where is restricted to the region outside the iron rod. Find this field. Express in terms of quantities given in the introduction (and constants), using the unit vectors in the cylindrical coordinate system, r^ ,theta^ , and z^ .