In Section \(9.1\), Faraday's law was used to show that the field \(\mathbf{E}=-\frac{1}{2} k B_{0} e^{k t} \rho \mathbf{a}_{\phi}\) results from the changing magnetic field \(\mathbf{B}=B_{0} e^{k t} \mathbf{a}_{z}\). (a) Show that these fields do not satisfy Maxwell's other curl equation. (b) If we let \(B_{0}=1 \mathrm{~T}\) and \(k=10^{6} s^{-1}\), we are establishing a fairly large magnetic flux density in \(1 \mu\) s. Use the \(\nabla \times \mathbf{H}\) equation to show that the rate at which \(B_{z}\) should (but does not) change with \(\rho\) is only about \(5 \times 10^{-6} \mathrm{~T}\) per meter in free space at \(t=0\).

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how electric currents and magnetic fields interact. It states that a changing magnetic field within a closed loop induces an electromotive force (EMF) which in turn causes an electric current if the circuit is closed. Mathematically, Faraday's Law is expressed as

\[ \text{EMF} = - \frac{\text{d}\Phi_B}{\text{dt}} \]
where \( \Phi_B \) represents the magnetic flux, which is the product of the magnetic field strength times the area through which the field lines pass, and the angle between the magnetic field lines and the normal to the surface. The key takeaway from Faraday's Law is that it's the changing magnetic field that creates an electric potential, not a static field. When applied in exercises, as seen with the electric field \( \mathbf{E} \) and magnetic field \( \mathbf{B} \) given in the original exercise, Faraday's Law often involves calculating the induced EMF or using the law to predict the behavior of electromagnetic fields in dynamic situations.

Magnetic flux density, commonly represented by the symbol \( \mathbf{B} \), is a measure of the strength and direction of the magnetic field in a given area. It is an essential concept in understanding how magnetic fields affect matter and is central to the formulation of Maxwell's equations. The magnitude of the magnetic flux density can be thought of as how many magnetic field lines are passing through a certain area, with the units of tesla (T) in the International System of Units (SI).

The exercise presents a scenario where a magnetic field changes exponentially with time, as given by \( \mathbf{B} = B_0 e^{kt} \mathbf{a}_z \). Here, the constant \( B_0 \) represents the initial magnetic flux density, while \( k \) depicts the rate of change with respect to time. This particular form of magnetic flux density highlights its dynamic nature and the direct impact it can have on the electric field, as governed by Faraday's Law. The concept becomes even clearer when, through the exercise solution, we observe the failure of the provided fields to satisfy Maxwell's other curl equation, underscoring the intricate relationship between electric and magnetic fields.

Cylindrical coordinates provide a systematic way of describing the position of a point in three-dimensional space. These coordinates are especially useful when dealing with problems that have cylindrical symmetry, such as those involving circular wires or cylindrical magnets.

In cylindrical coordinates, a point is represented by a trio of numbers \( (\rho, \phi, z) \), where \( \rho \) is the radial distance from the origin to the projection of the point onto the plane, \( \phi \) is the angle measured from a reference axis to the line connecting the origin to the point's projection, and \( z \) is the height of the point above the plane. The exercise demonstrates the use of cylindrical coordinates in calculating the curl of the electric field, \( abla \times \mathbf{E} \), a key operation in electromagnetism that shows the twisting or rotation of the field. The step-by-step solution employs the determinant form of the curl operator in cylindrical coordinates to verify Maxwell's equations. In doing so, it illuminates how electromagnetism can be explored from different coordinate perspectives, each best suited to the symmetries of the physical situation in question.