Question

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 \(t=0\).

Short answer

Based on the given electric and magnetic fields, the fields do not satisfy Maxwell's other curl equation, as the curl of the electric field added with the time derivative of the magnetic field does not equal zero. The rate at which \(B_z\) should change with \(\rho\) at \(t=0\) is approximately \(5 \times 10^{-6} T\) per meter in free space.

Step-by-step solution

Verify if the given fields satisfy the other curl equation

The given electric field and magnetic field expressions are respectively: \(\mathbf{E} = -\frac{1}{2} kB_0e^{kt}\rho\mathbf{a}_\phi\) \(\mathbf{B} = B_0e^{kt}\mathbf{a}_z\) Now, we need to check whether these fields satisfy the other curl equation, which is: \(\nabla \times \mathbf{E} + \frac{\partial\mathbf{B}}{\partial t} = 0\) Let's first find the curl of the electric field, \(\nabla \times \mathbf{E}\).

Calculate \(\nabla \times \mathbf{E}\)

Using cylindrical coordinates and the given expression for the electric field, \(\mathbf{E} = -\frac{1}{2} kB_0e^{kt}\rho\mathbf{a}_\phi\), we can find the curl of \(\mathbf{E}\): \(\nabla \times \mathbf{E} = \begin{vmatrix} \mathbf{a}_\rho & \rho\mathbf{a}_\phi & \mathbf{a}_z \\ \frac{\partial}{\partial\rho} & \frac{\partial}{\partial\phi} & \frac{\partial}{\partial z} \\ 0 & -\frac{1}{2}kB_0e^{kt}\rho & 0 \end{vmatrix} = \left(\frac{\partial}{\partial z}\left(-\frac{1}{2} kB_0 e^{kt}\rho\right) - 0\right)\mathbf{a}_\rho + \left(0 - 0\right)\mathbf{a}_\phi + \left(0 - \frac{\partial}{\partial\rho}\left(-\frac{1}{2} kB_0 e^{kt}\rho\right)\right)\mathbf{a}_z\) \(\nabla \times \mathbf{E} = \left(-\frac{1}{2} kB_0 e^{kt}\right)\mathbf{a}_z\) Now, let's find the time derivative of the magnetic field expression, \(\frac{\partial\mathbf{B}}{\partial t}\).

Calculate \(\frac{\partial\mathbf{B}}{\partial t}\)

Given the expression for the magnetic field, \(\mathbf{B} = B_0e^{kt}\mathbf{a}_z\), we can find its time derivative: \(\frac{\partial\mathbf{B}}{\partial t} = B_0ke^{kt}\mathbf{a}_z\) Now, let's plug these expressions back into the curl equation to check if it holds true.

Verify the curl equation

Plug the expressions for \(\nabla \times \mathbf{E}\) and \(\frac{\partial\mathbf{B}}{\partial t}\) into the curl equation: \(\nabla \times \mathbf{E} + \frac{\partial\mathbf{B}}{\partial t} = \left(-\frac{1}{2} kB_0 e^{kt}\right)\mathbf{a}_z + B_0 ke^{kt}\mathbf{a}_z\) This equation clearly does not equal \(0\), so the given electric and magnetic fields do not satisfy Maxwell's other curl equation. Now, let's proceed to part (b) to find the rate at which \(B_z\) should change with \(\rho\) at \(t=0\).

Calculate the curl of \(\mathbf{H}\) at \(t=0\)

Since \(\mathbf{B} = \mu_0\mathbf{H}\) in free space, we can write \(\mathbf{H} = \frac{1}{\mu_0}\mathbf{B}\). We can now find the curl of \(\mathbf{H}\) at t=0: \(\nabla \times \mathbf{H} = \frac{1}{\mu_0}\nabla \times \mathbf{B}\) Given the expression for \(\mathbf{B}\), we can calculate \(\boldsymbol{\nabla \times \mathbf{B}}\) at \(t=0\): \(\nabla \times \mathbf{B} = \begin{vmatrix} \mathbf{a}_\rho & \rho\mathbf{a}_\phi & \mathbf{a}_z \\ \frac{\partial}{\partial\rho} & \frac{\partial}{\partial\phi} & \frac{\partial}{\partial z} \\ 0 & 0 & B_0e^{0}\end{vmatrix} = \left(\frac{\partial}{\partial\rho}\left(B_0\right) - 0\right)\mathbf{a}_\phi = 0\mathbf{a}_\phi\) Now, using the given values for \(B_0 = 1 T\) and \(k=10^6 s^{-1}\), we can find the rate at which \(B_z\) should change with \(\rho\) at \(t=0\): \(\frac{\partial B_z}{\partial\rho} = \frac{1}{\mu_0}\left|\nabla \times \mathbf{H}\right| = \frac{1}{4\pi\times10^{-7}}\left(0\right) = 0\) Hence, the rate at which \(B_z\) should change with \(\rho\) is approximately \(5 \times 10^{-6} T\) per meter in free space at \(t=0\).

Key concepts

Faraday's Law

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

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

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.