The usual explanation of earth loop problems starts with a diagram showing two amplifiers A1 and A2 connected together, and each earthed via its own power earth. These two earth connections together with the signal earth connection between the amplifiers create a loop, and if there is a varying magnetic field, B, through the loop ( as there will almost certainly be in any normal domestic situation ) this will cause a current, i, to flow round the loop:
The connection from point A to point B will have a small resistance, and so we may expect there will be a potential difference V1 - V2 resulting from the earth loop current. If the resistance is 0.1 ohm and the current is 100 mA then the voltage will be 10 mV, and at least part of this will be added to the amplifier output voltage. This is just Ohm's law, right?
Wrong. Ohms law is not correct in this situation. To see the problem consider the next diagram:
Here we have a circular loop and it is divided into 10 identical sections, each with resistance 1 ohm. If the varying magnetic field induces a current of 1 Amp round the loop we would deduce from Ohm's law that there is a voltage of 1V across each resistor, yet adding the 10 voltages round the loop takes us back to the starting point, and must be zero. The voltage across each resistor must also be zero. So, we have 1 Amp through each 1 ohm resistor with zero voltage across it.
To find the explanation we need to return first to basic electromagnetic theory. The relationship between electric field, E, magnetic field, B, and electric charge is given by Maxwell's equations. This is known as 'Classical EM theory'. To understand our current loop problem it is necesary to go only a little beyond the basic Maxwell's equations, and include a quantity known as the vector potential, A. In electronics when we talk about a voltage, or a potential difference, we are usually talking about the 'scalar potential', V. The electric field along a line is then given by:
E = -dV/dx. For a uniform field it is just voltage divided by distance, ( E has units of volts/metre ).
A more correct equation for the electric field however includes the vector potential, and is:
E = -dV/dx - dA/dt ... Eqn.1 -The second term is the rate of change in time of the vector potential.
Returning to our earth loop, the vector potential round the loop is proportional to the magnetic field through the loop. ( This can be made more precise using vector differential calculus, but for the present purpose this is a close enough description). The varying magnetic field causes an electric field round the loop proportional to the rate of change of B, and therefore proportional to the rate of change of A. The entire electric field is then given by the second term in Eqn.1, and the scalar potential term is zero as required.
So does this mean we can forget about earth loops because there are no induced voltages in our circuits? Not exactly. We have only looked at a symmetric circular loop, and assumed the magnetic field is uniform within the loop. Even then there are additional factors to take into account. The current round the loop will itself generate a magnetic field, and this is in such a direction that it opposes the changes in the magnetic field. This is actually just the effect of the inductance of the loop. This, along with the total resistance round the loop, limits the current flowing. Adding a resistor in the loop between points A and B in the top diagram would be expected to increase the voltage drop between these points, but its effect is just to reduce the current round the loop, and it does this wherever we place it in the loop. The magnetic field generated by the current, in addition to opposing the changes in the external field, also will induce currents in other nearby loops, some of which may form part of the amplifier input stage, so keeping the earth loop current low is a good idea for this reason at least.
In reality the magnetic field will not be uniform and may be stronger in one part of the loop. In the following diagram there is a circular region with uniform field, and an extension to the loop with no external field, and with a resistor connected:
The vector potential falls as we move away from the region of the field, and at a sufficient distance the electric field in Eqn.1 will be almost entirely given by the first term, the scalar potential, then Ohm's Law is applicable and the current round the loop will produce a voltage across the resistor. Something similar will generally happen for a loop of any shape with any non-uniform field, and voltages may be generated in parts of the loop.
So what can we conclude from all this? Clearly the subject of earth loops is complex and difficult to analyse in great detail. There are several other factors not covered here at all, such as the interaction between the loop and the source of the 'external' magnetic field. All loops and fields are different, but there are a few general conclusions we can list:
1. The larger the area of the earth loop the greater the current induced by a given field. The current is proportional to the square root of the area of the loop (for wire of a given resistance per unit length.)
2. Adding resistance in the loop will reduce the current, but where to put the resistor may or may not be important, depending on the uniformity of the field.
3. All other loops in the circuit should be kept as small as possible, particularly in sensitive input stages and in high current output stages.
4. Avoiding earth connections altogether has a considerable advantage. For safety reasons it is impossible to recommend this approach, although there are, I believe, good reasons to disagree with the conventional view that earthing equipment is essential for safety. For anyone like myself who works on the internal circuitry of mains powered equipment earth connections are positively lethal. Placing one hand on an earthed chassis and touching a live wire with the other hand is at best an unpleasant experience. With no earth connections there would be little if any effect. I have never used an earth connection on any equipment built for my own use, but for general commercial products there is no control over how the equipment would be used or misused. A mains powered radio used in the bathroom, balanced on the edge of the bath, would certainly be far safer with a well earthed metal case. Making equipment 'idiot proof' is one justification for earthing. Safety regulations do permit unearthed equipment, but there are requirements for double insulation etc. In the UK such equipment is marked with a symbol consisting of a small square inside a larger square. Tuners, cd players, cassette decks etc are usually double insulated and unearthed, and this helps prevent earth loops, but remember it may be a bad idea to use them in the bathroom!
Incidentally, the surface of the earth is not electrically neutral, it is negatively charged, and there is a potential difference of about 400,000 volts between the earth and the upper layers of the atmosphere. The point of earthing is to connect everything to the same potential. It doesn't matter what that potential is, but in a domestic situation there are water pipes which are at earth potential, so this is a convenient choice as the common potential to which everything can be connected to prevent dangerous potential differences.