Adding Electric Fields - Find Simplification Solution

• PhDorBust
In summary, the conversation discusses simplifying two given electric fields into one magnitude and exponential form. The process involves factoring out a common factor, using Euler's formula, and converting from cartesian to polar coordinates in the complex plane.
PhDorBust

Homework Statement

Have two electric fields.

$$\hat{x} E_1 e^{i(kz- \omega t)}$$
$$\hat{x} E_2 e^{i(-kz- \omega t)}$$

Where E_1, E_2 are real.

Sum them such that the result can be expressed as one magnitude and exponential, e.g., |E|e^(iq), Where E is real.

I have no clue how I would begin to simplify this. Any ideas? It's from an undergraduate text.

Factor out the common factor of e-iωt. Use Euler's formula, e=cos θ + i sin θ.

Sure.. but that's trivial detail and not key step.

$$e^{-i \omega t}[(E_1 + E_2) cos(kz) + i(E_1 - E_2) sin(kz)]$$

Well, the rest is even more trivial, so I have no idea where you're getting stuck.

Perhaps wording is unclear. Here is answer. Looks little complex for name of trivial, so long that I attach rather than typeset in latex =].

Disregard the unit vector.

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PhDorBust said:
Sure.. but that's trivial detail and not key step.

$$e^{-i \omega t}[(E_1 + E_2) cos(kz) + i(E_1 - E_2) sin(kz)]$$

So now, how do you calculate the absolute value of a complex number? And its argument? That's just converting from cartesian to polar coordinates in the complex plane (if we ignore the factor e^(-iωt) which you can set apart).

(The only non-trivial part apart from that conversion will be using the identity sin² φ + cos² φ = 1).

What is an electric field?

An electric field is a physical field produced by electrically charged objects. It exerts a force on other charged objects within its vicinity.

How do you add electric fields?

To add electric fields, you need to calculate the vector sum of the individual electric fields. This involves adding the magnitudes and directions of each field vector to find the overall electric field at a specific point.

What is the purpose of finding a simplification solution for electric fields?

Finding a simplification solution for electric fields allows us to better understand and analyze the behavior of electrically charged particles. It also helps to simplify complex systems and make calculations more manageable.

What are some common simplification techniques for adding electric fields?

Some common simplification techniques for adding electric fields include using superposition, symmetry, and the principle of superposition. These techniques can help to reduce the complexity of the problem and make calculations easier.

What are some applications of adding electric fields?

Adding electric fields is used in various fields such as physics, engineering, and electronics. It is essential for understanding the behavior of electrically charged particles in electric circuits, motors, and other devices. It is also used in studying the behavior of charged particles in natural phenomena such as lightning and auroras.

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