Forces between two moving charges
Now let the charges move to the left with velocity v (sketch at right). Each moving charge is like a small element of electric current.
Like electricity*, the magnetic interaction is an inverse square law, and the law of Biot-Savart gives the field B at distance r due to a small length dL carrying current I. For this geometry, the magnetic field at one charge due to the presence of the other is
where kmag is Coulomb's constant for magnetism, μ0 is the magnetic permeability of space and where the direction is perpendicular both to r and to v (out of the page, in this instance, for the field due to the upper charge at the position of the lower, if q>0).
* Some people are surprised by this: expressions for electric field due to charges usually have the factor 1/r2 in them, whereas those for magnetic field due to a current usually have factor 1/r. The difference is due to the usual geometry: currents usually flow in long thin wires (approximately one-dimensional things). Charges may be very small (approximately zero dimensional things). Calculating the field due to a one-dimensional distribution involves an integration with respect to position along the wire, which usually removes a factor of r from the denominator. Note also that, in the Coulomb version for both forces, there is no 4π. Whether or not it appears depends on the definition of the constant. Aesthetically, it is appealing to have 4π precede r2: In this formulation, Gauss' laws make it clear that the inverse square law (in both cases) is a statement about the geometry of the universe, rather than some special property of electromagnetism.
In this case, if the charge q covers the small distance dL in time dt, then IdL may be replaced with qdL/dt = qv. In this case, the field and the velocity of the second charge are at right angles, so the force on the second charge has the magnitude Bqv, which is here attractive and
, so
Total force between two moving charges; rearranging
We’ll see later that the magnetic force is usually quite small compared with the electrostatic force. So let's treat this small magnetic term as a modification or correction to the electric term and write an expression for the nett repulsive force like this:
, i.e.
Now the term in parentheses is a pure number, so we can see that (kelec/kmag) or (1/ε0μ0) must have the units of speed squared. So let's write the preceding equation as:
This speed v0 is a critical value in this problem: if our two charges were travelling at this speed, then the force you would calculate between the charges using this equation would be zero: the magnetic attraction would be equal to the electric repulsion, but opposite.
So what is v0? If you have studied waves, the format of this expression may seem familiar. The speed of a wave is given by:
For example, the speed of a wave in a string and the speed of sound in a medium are respectively:
where F is the tension in the string, μ is its mass per unit length and where κ is the adiabatic bulk modulus of elasticity and ρ the density of the medium through which sound travels. κ is the ratio of the pressure difference applied to the fractional reduction in volume that it produces (and, in air, it is about 1.4 times the pressure of the air). Note that F and κ represent the property that returns a displacement of the string or the medium to equilibrium, and that the one and three dimensional densities μ and ρ represent the mass and thus the inertia, the capacity of the medium to overshoot the position of equilibrium. Both properties are required for an oscillation and for the transmission of a wave.
Here, the medium for the wave is vacuum (and we might expect that the wave is electromagnetic). In a sense, kelec represents its elastic or restoring property, because the electric field acts to move the charges in directions that reduce the volume over which the field acts. In the same metaphorical way, kmag is analogous to inertia: Faraday's law tells us that a magnetic field, once established, tends to produce electrical effects that act to maintain the magnetic field.
Alternatively, we can go straight to Maxwell's equations, which tell us almost directly that the speed v0 that appeared in our equestion above is the speed of light and other electromagnetic waves:
So our equation for the total force between the two charges is:
In Special Relativity, the factor γ (or its reciprocal) appears regularly as the factor that relates physical measurements made in two inertial frames with relative speed v. In this case, the motion of the two charges reduces this force between them by a factor of γ2. Here, an observer travelling at v to the left sees just the electric repulsion. Another observer, at rest on the page sees electrostatic force minus magnetic force. So (as we’ve shown in this example), magnetism is the relativistic correction for electrostatics, when considering moving charges.
This naïve analysis does give some insight into the relation between electricity and magnetism, though we should also apply the Lorentz transforms and check that relativity factor. More insight about magnetism comes from the symmetry between electricity and magnetism in Maxwell's equations. Further, in relativity, electromagnetic fields are often treated as vectors with six components, and that's what we should do here for a complete analysis. But that is a whole new story and we have another question to answer:s
So why isn't magnetism negligible?
My first reaction to the analysis above was surprise. We usually think of γ as a term that is very close to unity, except for velocities close to c. Yet currents with electron drift velocities very much smaller than c can produce large magnetic forces: for example, think of the electric motors that drive trains. How could a relativistic term concerning such low electron drift velocities produce the large forces that accelerate trains?
Let me answer with an analogous question: how could gravity, which is such a puny force, govern the universe? The ratio of the electrical attraction between a proton and an electron to the gravitational attractions between them is 2×1039. Why isn't gravity negligible? The answer is, of course, that while there is no negative mass, there is negative charge. Consequently, on any large scale, electrical forces cancel out and gravity doesn't. In our case, if we removed the conduction electrons from two parallel wires (and somehow stopped the huge concentration of the remaining ions from exploding!), the electrostatic forces exerted between the ions in those two wires would be stupendous. However, thse forces are approximately neutralised by the forces exerted betwen the equal negative charges carried by electrons.
Approximately, but not quite. At least, not if there are currents flowing. When currents flow, the repulsive forces between the moving electrons are reduced by a factor that is very, very slightly less than one: ie by (1/γ2). So, in the laboratory frame, we might say that the repulsive forces between the electrons in one wire and those in the other, plus that between the protons in the two wires, is very slightly less than the attractive force between the protons and the electrons. The imbalance between the two is then the magnetic force. The geometry and the mechanics involved in magnetic atoms (and thus in bar magnets) are more complicated. But that ‘magical’ action-at-a-distance that delighted my infant self as I played with magnets is indeed one of the effects of special relativity.
Okay, but what is electricity?
My argument above aims to explain magnetism in terms of electricity. There’s no comparable way of explaining electricity in terms of something more fundamental – the electrical interaction really is fundamental and can’t be explained in terms of something more basic.
Is there anything more to say? We could apply the anthropic argument to electricity: Without the electric interaction there would be no chemistry. Without chemistry, there would be no humans to wonder why there is an electric interaction. Some cosmologists propose multiple universes, some or many of which might have different laws of physics. If they’re right, it’s no surprise that we find ourselves in a universe that allows chemistry and chemistry-based lifeforms.
We might also observe that electricity is very simple – it’s almost the simplest form of interaction and so an admirable candidate for being fundamental. As we mentioned above, the 1/r2 is a property of three-dimensional space (shared with gravity, magnetism, light intensity and other features) rather than a special feature of electricity. Having positive and negative charges makes it look slightly more complicated than (Newtonian) gravity. But that feature is fundamental for chemistry.
So electricity is basic in both senses. But for magnetism, there is an answer to the question at the top of this page.
I originally posted this story as The electric and magnetic forces between moving charges on the site Einstein Light which we made to celebrate the centenary of Special Relativity.
This page examines magnetic (and electrical) forces in a very simple geometry and rearranges the expression in a way that explains magnetism.
To analyse it, let's start by simplyfying and set the velocity equal to zero. Two stationary charges q at separation r, as in the sketch at right.
