I have recently been studying waves and have been usually using equations of the form $A\mathrm{exp}(ikr-ωt)$. However , as I was going through my notes I noticed a different expression for ELECTROMAGENTIC WAVES , which was $E\mathrm{exp}(ikr-ωt)$ and $E$ was defined as the electric field. My question is : Is this how we define an electric field ? As the amplitude of an electromagnetic wave ?
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$\begingroup$ The definition of the electric field is "a field describing the force that a charged particle would experience due to electrical interactions at each point in space" (to be more comprehensive we should talk about the electromagnetic field and electromagnetic interactions). A sinusoidal wave is one particular behavior we can see in the electric field, but it is not the definition of the electric field. Whether the prefactor in the formula for a sinusoidal wave is called "A" or "E" has no bearing on the actual physics. $\endgroup$– The PhotonCommented Mar 9, 2018 at 17:26
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"[H]ow [do] we define an electric field ? As the amplitude of an electromagnetic wave ?"
More nearly the other way round…
An electric field is a region in which a charged particle, whether stationary or moving, experiences a force. That's pretty much the definition of an electric field. The electric field strength, $\vec{E}$, at a point P is defined as$$\vec{E}=\frac{\text{force on a 'test' charge placed (temporarily) at P}}{\text{charge of test charge}}$$
Stationary or moving charges are sources of electric fields. Moving charges are also sources of magnetic fields. Accelerating charges are sources of electromagnetic (e-m) waves. These consist of electric and magnetic fields travelling together as waves, that is the fields obey wave equations. They can travel through a vacuum, where there are no particles of matter, and all we have to describe the e-m wave are the fields themselves. We usually measure the amplitude of the wave by the maximum value of its electric field.
answered Mar 9, 2018 at 18:27