I have heard, (I can't recall where), that the Earth's natural magnetic field is slowly declining. I have also heard that solar flares can disrupt the operation of electrical/electronic equipment. Would either of these natural phenomena affect the operation of electronic/electrical equipment? Please answer each query briefly, and provide links if possible.
Electrical/Electronic equipment is only effected by something called Electro Magnetic Interference (EMI). The Earth Magnetic field usually does not create this kind of interference. The solar flares on the other hand can produce EMI. However, there are ways you can protect your electrical equipment from the EMI by shielding the equipment from the source of EMI. The thickness of the shield depends on the frequency of the EMI and the conductivity of the metal that is using as the shield. In general, a good conductor metal is the good shield such as Nickel, Silver, Copper, etc. Another way to protect a electrical component against the EMI is using a high permeability metal (hi-mu metal).
The magnetic field of the Earth is only an issue with things that measure or otherwise require the presence of the field, specifically compasses, only at that, only magnetic compasses. Most modern navigational equipment use gyrocompasses, which measure the rotation axis of the Earth and therefore, do not care whether a magnetic field exists. If the magnetic field were to disappear tomorrow, only magnetic compasses and certain geologists would be affected.
