Is there any example where electric and magnetic fields are not perpendicular? Perpendicular electric and magnetic field creates light or other electromagnetic waves. Is it a necessary property to have a perpendicular fields? If not what would happen when the fields are not perpendicular?
 A: 1) It is necessary for a plane EM wave. 
If one assumes solutions to the Maxwell's equation to be plane waves, it is not hard to show that $\vec B \cdot \vec E = 0$.  
Namely, take the third Maxwell's equation and dot both sides with $\vec E$.
$$\nabla  \times \vec E =  - {{\partial \vec B} \over {\partial t}}$$
$$i\vec k \times \vec E = i\omega \vec B{\rm{ }}$$
$$(\vec k \times \vec E) \cdot \vec E = \omega \vec B \cdot \vec E$$
$$0 = \vec B \cdot \vec E$$
2) In general case, it is not necessary.  
Assume a constant current flowing up along the z axis. It produces static magnetic field around it in the form of concentric circles (and no electric field as net charge inside the conductor is zero).  
Put a positive charge to a point (-1,0,0). It produces static electric field with radially outward lines. 
At the point (0,1,0) the angle between electric and magnetic fields is 45 degrees.

A: Yes there is! In wave guides, there are modes of propagation. What we know as transverse electromagnetic field is just the $TEM$ mode of a field. There are also the $TE$ (transverse electric mode) with $E_z≠0$ and the $TM$ (transverse Magnetic mode) with $B_z≠0$ where $z$ is the direction of propagation!
A: Electricity can be static, like the energy that can make your hair stand on end. Magnetism can also be static, as it is in a refrigerator magnet. A changing magnetic field will induce a changing electric field and vice-versa—the two are linked. These changing fields form electromagnetic waves. Electromagnetic waves are formed by the vibrations of electric and magnetic fields. These fields are perpendicular to one another in the direction the wave is travelling. Once formed, this energy travels at the speed of light until further interaction with matter. 
Maxwell developed a scientific theory to explain electromagnetic waves. He noticed that electrical fields and magnetic fields can couple together to form electromagnetic waves. The terms light, electromagnetic waves, and radiation all refer to the same physical phenomenon: electromagnetic energy. This energy can be described by frequency, wavelength, or energy. All three are related mathematically such that if you know one, you can calculate the other two. There’s a particle model and a wave model for electromagnetic radiation, and as we know electromagnetic radiation have dual nature. For an electromagnetic radiation to persist the two electric and magnetic fields have to propagate in a perpendicular direction to each other, EM waves consist of oscillating electric and magnetic fields.
A: Electric and magnetic fields are always perpendicular to each other
A: According to my understanding at physics, an electric field which is static cannot create any magnetic field.According to faraday for a magnetic field to exist there should be a varying electric filed(which varies over time).A simple example is battery connected to coil which is moved in and out near the galvanometer shows deflection(A galvanometer is device which detects magnetic filed by coupling magnetic filed near its vicinity to the coil inside and it internally has magnet which orients along the direction of earths's.The needle which is connected the coil coupling the magnetic field experiences a torque due to the EMF induced in it,and rests in the direction such that the earths magnetic field and the coupled magnetic are perpendicular to each other..hence the deflection of the galvanometer needle shows the stregth of magnetic field.
