Maxwell's Equations: Draw State of Polarization & Find B(x,t)

• leoflindall
In summary, the conversation involved discussing the state of polarization of an electromagnetic wave and how to use Faraday's law to obtain the magnetic field from the electric field. The electric field was found to be circularly polarized (anticlockwise) with an amplitude of E0, and the use of Faraday's law was explained to solve for the magnetic field. There was also a correction made to the use of real components.
leoflindall

Homework Statement

Draw the state of polarization of the electromagnetic (EM) wave defined by

*****PLEASE NOTE EQUATION SHOWN IN NEXT POST****** (For some reason can't change it in this post...

with Eo real. Use a sentence to describe in words the state of polarization of this EM wave.

Use the differential form of Faraday's law to obtain B(x,t) for the same EM wave

Homework Equations

*****PLEASE NOTE EQUATION SHOWN IN NEXT POST****** (For some reason can't change it in this post...

The Attempt at a Solution

I have done the first part and found the EM wave to be circularly polarized (anticlockwise) with an amplitude of E0

i am unsure how to use faradays law. I tried breaking the LHS of faraday into its respective partial differential vector form, which i am currently working my way through but i am fairly sure it is wrong!

Any ideas on how to apply to get the magnetic field from the electric using faraday-maxwell's law would be greatly appreciated!

Many Thanks

Last edited:
Sorry i copied and pasted the equation and it has come out wrong...

E(x,t) = Eo( y + e^(i3$$\Pi$$/2) z )e^(i(kx-$$\omega$$t)) ,

Bold Letters denote unit vectors

Relevant Equations;

$$\nabla$$ x E = - $$\partial$$B / $$\partial$$ t (Faraday-Maxwell Equation)

Excuse my mistake!

Last edited:
Your electric field is given by

$$\mathbf{E}=E(x,y,z,t)=E_0\exp[i(kx-\omega t)]\hat{\mathbf{y}}+E_0\exp[i(kx-\omega t)]\exp\left[i\frac{3\pi}{2}\right]\hat{\mathbf{z}}$$

correct? But we also know $\mathbf{E}=Re(\mathbf{E}'\exp[i\omega t])$ where $\mathbf{E}'$ is the spatial component of the electric field. Then through Faraday's law,

$$\mathbf{B}'=-\frac{1}{i\omega}\nabla\times\mathbf{E}'$$

You can then solve $\mathbf{B}=Re(\mathbf{B}'\exp[i\omega t])$.

Now that I've had a good night's rest, you can actually ignore the fact that I said the real components of the spatial electric and magnetic components. This would only be true if your original electric field were given by cosine.

Thank you, that really helps! I appreciate it!

1. What are Maxwell's Equations?

Maxwell's Equations are a set of four equations that describe the relationship between electric and magnetic fields in a region of space. They were developed by James Clerk Maxwell in the 19th century and are fundamental to the study of electromagnetism.

2. How do Maxwell's Equations relate to polarization?

One of Maxwell's Equations, known as the Maxwell-Ampere Law, describes how changing electric fields can generate magnetic fields. This is important in understanding the behavior of polarized light, which is created when light waves have a specific direction of oscillation in the electric field.

3. What is the state of polarization?

The state of polarization refers to the direction and magnitude of the electric field oscillations in a light wave. It can be linear, circular, or elliptical, depending on the orientation and shape of the electric field vector.

4. How can I draw the state of polarization?

The state of polarization can be represented graphically using a polarization diagram, which shows the direction and magnitude of the electric field vector in relation to the direction of propagation of the light wave. It can also be represented using a Poincaré sphere, which is a three-dimensional representation of the polarization state.

5. How do I find B(x,t) using Maxwell's Equations?

To find B(x,t), we can use the Maxwell-Faraday Law, which states that changing magnetic fields can induce electric fields. By solving this equation along with the other three Maxwell's Equations, we can determine the behavior of both electric and magnetic fields in a region of space.

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