Question

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields $$\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i} \quad \mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}$$ satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

Short answer

Explain why, for the given plane wave, the curl of the magnetic field yields only the x-component, and not the other components. For the given plane wave, the magnetic field \(\mathbf{B}(z, t) = A \sin (kz - \omega t) \mathbf{j}\) only has a non-zero component in the y-direction. To find the curl of the magnetic field, we use the formula: $$\nabla \times \mathbf{B} = \left(\frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z}\right)\mathbf{i} - \left(\frac{\partial B_z}{\partial x} - \frac{\partial B_x}{\partial z}\right)\mathbf{j} + \left(\frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}\right)\mathbf{k}$$ Given the magnetic field has only a y-component (\(B_x = 0\) and \(B_z = 0\)), most of the terms in the curl formula become zero. The only non-zero term is the \(-\frac{\partial B_y}{\partial z}\) term, which is associated with the x-component (\(\mathbf{i}\)). Therefore, only the x-component will be non-zero when calculating the curl of the given magnetic field.

Step-by-step solution

Calculate the curl of the magnetic field \(\nabla \times \mathbf{B}\)

To calculate the curl of the magnetic field, we will use the curl definition in Cartesian coordinates, which is given by: $$\nabla \times \mathbf{B} = \left(\frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z}\right)\mathbf{i} - \left(\frac{\partial B_z}{\partial x} - \frac{\partial B_x}{\partial z}\right)\mathbf{j} + \left(\frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}\right)\mathbf{k}$$ Given the magnetic field vector \(\mathbf{B}(z, t) = A \sin (kz - \omega t) \mathbf{j}\), we can observe that \(B_x = 0\), \(B_y = A \sin (kz - \omega t)\) and \(B_z = 0\). Thus, the curl becomes: $$\nabla \times \mathbf{B} = - kA \cos(kz - \omega t) \mathbf{i}$$

Calculate the time derivative of the electric field \(\frac{\partial \mathbf{E}}{\partial t}\)

Now, we need to calculate the time derivative of the given electric field, \(\mathbf{E}(z,t) = A\sin(kz-\omega t)\mathbf{i}\). The time derivative is: $$\frac{\partial \mathbf{E}}{\partial t} = -A\omega\cos(kz - \omega t)\mathbf{i}$$

Show that the equation is satisfied if \(\omega=k/C\)

We can now compare the two sides of Maxwell's equation, \(\nabla \times \mathbf{B}\) and \(C\frac{\partial \mathbf{E}}{\partial t}\). In order for this equation to be satisfied, the two vectors must be equal: $$\nabla \times \mathbf{B} = C \frac{\partial \mathbf{E}}{\partial t}$$ Substituting our previous results: $$-kA \cos(kz - \omega t) \mathbf{i} = C(-A\omega\cos(kz - \omega t)\mathbf{i})$$ We can see that the equation is satisfied if \(\omega=k/C\).

Sketch the directions of the electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\)

The electric field \(\mathbf{E}(z,t)\) lies along the \(\mathbf{i}\) direction (x-axis), and the magnetic field \(\mathbf{B}(z,t)\) lies along the \(\mathbf{j}\) direction (y-axis). Since both fields are sinusoidal with the same phase, they will be in phase with each other, oscillating between the positive and negative xy-plane quadrants. A rough sketch should show the electric field in the x-axis direction and the magnetic field in the y-axis direction. Both fields are perpendicular to each other, and they oscillate in phase.

Key concepts

Electromagnetic Waves

Electromagnetic waves are fascinating oscillations composed of electric and magnetic fields that travel through space. These fields oscillate perpendicular to each other and to the direction of wave propagation. This unique behavior of electromagnetic waves is dictated by Maxwell's equations, which describe how electric and magnetic fields interact and propagate.
Electromagnetic waves include a wide range of radiations such as visible light, radio waves, and X-rays. Each of these types of waves has different frequencies and wavelengths, which in turn defines their energy and interactions with matter. A key aspect of understanding electromagnetic waves is recognizing that they do not require a medium to travel. They can propagate through the vacuum of space, making them fundamental to transmitting signals across the universe.

• Electromagnetic wave equation: - Relates oscillating electric field \(\mathbf{E}\) to magnetic field \(\mathbf{B}\) - Wave propagates at a speed determined by the medium (in vacuum, it travels at the speed of light,\( c \approx 3 \times 10^8\, m/s\)).
• They consist of two sinusoidally varying fields: electric field \(\mathbf{E}(z, t)\) and magnetic field \(\mathbf{B}(z, t)\).

Understanding electromagnetic waves is crucial not only for physics but also for technologies we use daily, such as wireless communication and remote sensing.

Vector Calculus

Vector calculus is essential in studying electromagnetic fields and waves. It involves various differential and integral operations used to analyze vector fields, which are fundamental in describing the behavior of electromagnetic fields.
One key operation in vector calculus is the curl, which measures the rotational tendency at a point in a vector field. In the context of electromagnetic waves, the curl of the magnetic field \(abla \times \mathbf{B}\) is linked to the changing electric field over time, as seen in Maxwell's equations. This curl operation helps in understanding how magnetic fields loop and play a role in wave propagation.

• Curl (abla \times \mathbf{F}): - Describes the rate of rotation or ''curling'' of a vector field \(\mathbf{F}\). - Used in computing the behavior of magnetic fields in Maxwell's equations.
• Divergence (abla \cdot \mathbf{F}): - Not used in this problem but essential in other Maxwell's equations.
• Vector calculus is powerful: - Simplifies complex problems. - Facilitates the understanding of physical phenomena such as electromagnetism.

Mastering vector calculus is crucial for anyone looking to excel in fields such as physics and engineering, particularly when analyzing electromagnetic phenomena.

Electric and Magnetic Fields

Electric and magnetic fields are central to the study of electromagnetism, directly impacting how electromagnetic waves behave. Electric fields \(\mathbf{E}\) exert forces on charged particles, while magnetic fields \(\mathbf{B}\) influence moving charges. Together, they form the backbone of Maxwell's equations.
In electromagnetic waves, electric and magnetic fields are always perpendicular to each other and to the wave's direction of travel. The strengths of these fields oscillate sinusoidally, meaning they follow a repeating pattern described by sine functions. This oscillation is what gives rise to the wave-like nature of electromagnetic waves.

• Electric Field, \(\mathbf{E}(z, t)\): - Described by a sine wave: \(\mathbf{E}(z, t) = A \sin(kz - \omega t) \mathbf{i}\). - Points in the x-axis (\(\mathbf{i}\) direction).
• Magnetic Field, \(\mathbf{B}(z, t)\): - Also described by a sine wave: \(\mathbf{B}(z, t) = A \sin(kz - \omega t) \mathbf{j}\). - Points in the y-axis (\(\mathbf{j}\) direction).

When considering these fields' directions and magnitudes, it's evident that they are interdependent, oscillating in harmony. This relationship underscores the principles laid out by Maxwell, revealing the unity and simplicity of electromagnetic theory.