Electromagnetism electric fields

In summary, the expression for an electric field in free space is given by E=Re{Eo exp[j(ωt-κy)\hat{i}, where Re represents the real part of a complex expression and j represents the imaginary unit. This is a plane wave propagating in the y direction and oscillating in the x direction. The transverse nature of the wave is determined by Maxwell's equations for free fields in vacuo, specifically the equation ∇⋅E=0. The exponential form can be converted to sine and cosine functions using Euler's relationships.
  • #1
james walshe
7
0
An Electric field has the following form in free space;

E=Re{Eo exp[j(ωt-κy)[itex]\hat{i}[/itex]

I am confused as to why a unit i vector is in the expression for an electric field oscillating in the Y direction? and also what does the Re mean ? i was reading somewhere about it being to do with rectangular coordinates.
 
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  • #2
Re is the real part of a complex expression, and [itex]\mathrm{j}[/itex] is used by electrical engineers as the imaginary unit, i.e., [itex]\mathrm{j}^2=-1[/itex].

A electromagnetic wave in free space must be transverse (both, the electric and magnetic components of the electromagnetic field are vectors perpendicular to the direction of the wave propagation). So what you have here is a plane wave
[tex]\vec{E}(t,\vec{x})=\left [\mathrm{Re} \; E_0 \cos(\omega t-k y)-\mathrm{Im} \; E_0 \sin(\omega t-k y) \right ] \vec{i}.[/tex]
This is a wave propagating in [itex]y[/itex] direction (if [itex]k>0[/itex]) and oscillating in [itex]x[/itex] direction).

That the wave must be transverse follows from Maxwell's equations for free fields in vacuo. Among them you have
[tex]\vec{\nabla} \cdot \vec{E}=0,[/tex]
and this is fulfilled for the above wave.
 
  • #3
Hi Vanhees71 thank you for the swift reply.
Did you use the Euler relationships to turn the exponential into the sine and cosine functions? i.e [e^jx=cosx+jsinx] and hence the imaginary part can be ignored? or can the expression be left in the exponential form.
 

1. What is electromagnetism?

Electromagnetism is a branch of physics that studies the relationship between electricity and magnetism. It explains how electric fields and magnetic fields are generated and interact with each other.

2. What is an electric field?

An electric field is a region in space where an electrically charged particle experiences a force. It is created by a charged particle and can exert a force on other charged particles within its vicinity.

3. How are electric fields created?

Electric fields are created by the presence of electric charges. Positive charges create electric fields that point away from them, while negative charges create electric fields that point towards them. The strength of the electric field is determined by the magnitude of the charge and the distance from the charge.

4. How do electric fields affect charged particles?

Charged particles in an electric field will experience a force, known as the electric force, that is proportional to the strength of the field and the magnitude of the charge. The direction of the force depends on the charge of the particle and the direction of the electric field.

5. What is the relationship between electric fields and magnetic fields?

Electric fields and magnetic fields are closely related and can even transform into each other. A changing electric field can create a magnetic field, and a changing magnetic field can create an electric field. This phenomenon is known as electromagnetic induction and is the basis for many modern technologies, such as generators and transformers.

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