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I've been having difficulty in understanding how an electromagnetic wave propagates through space.

Firstly, from what I understand, electromagnetic waves are made up of a constantly varying electric field placed perpendicular and in phase with a constantly varying magnetic field. Now lets set up a source to generate an electromagnetic wave. From the above, the source should consist of a varying electric field and varying magnetic field.

Now consider a point P a distance d away from the source. The time taken for the energy from the wave to reach the point P from the moment the source is switched on would then be

t = d/c.

Isn't the effect of a magnetic or electric field assumed to be instant before relativity showed that information cannot be transfered faster than the speed of light?

Even so, how does 2 perpendicular fields explain how energy is transfered through space? From what I've read from texts its because the electric field induces a magnetic field (Maxwell's law of induction), and the induced magnetic field induces yet another electric field (Faraday's law of induction) and so and so forth. But I failed to understand that as it still does not seem to explain how the wave propagates through space. The induced B-fields and E-fields are still induced in the space about the source, and I'm unable to see how it travels.

Perhaps someone could point me in the right direction?