![]() What happens between the two charges? Well, the magnetic field of points into the page at, so the magnetic force on is to the right, and the magnetic field of is out of the page at, so the magnetic force on is upward. The net electric force between the two charges is repulsive and opposite, but the magnetic forces aren't, so the electromagnetic force on on is equal but not opposite to the force of on, in violation of Newton's third law! We've got a problem, and we're going to solve it by invoking the momentum of the EM field. The way to recover conservation of momentum proceeds the same way we recovered the conservation of energy via the Poynting vector. Starting with the basic Coulomb/Lorentz laws, we'll write down an expression for the electromagnetic force on charges in a volume. We're going to integrate that over all space, which can have any distribution of charge, and relate that expression for force to an expression which only involves the field. In the interest of brevity, we'll skip around a bit and leave the full derivations for the real textbook. ![]() Suppose we have a volume containing some distribution of charge, current, and electromagnetic fields. ![]() It's handy to define the force per unit volume : The total force on that volume isĪgain, the goal is to replace anything that looks like a source in favor of fields, using Maxwell's equations. Similar to the derivation of the Poynting theorem, also using the other two Maxwell equations we haven't yet, we get Skipping through a few steps, we cut to the chase. Where is the Poynting vector and is the so-called "Maxwell stress tensor." To keep in mind what kind of units we're talking about here, has units force per unit volume, and the divergence will strip one spatial dimension, so the Maxwell stress tensor will have units of stress (force per unit area). For "pressure" forces, the force and area are in the same direction, and in the "shear" case the force and area are orthogonal.Īs we do the volume integral to go from to The tensor has diagonal "pressure" terms and off-diagonal "shear" terms. The first term on the right is related to the momentum stored in the electromagnetic field. We identify another useful term as the first integrand on the right: The second term is the rate at which momentum flows across the surface, and we describe the left-hand-side as the rate of change of the momentum of charges within the volume. Which is the momentum density within the fields. Just as a note, the signs here are swapped from the Poynting theorem - the Maxwell stress tensor is defined such that momentum flowing into the region corresponds with increasing, and vice-versa, opposite the case we had with. Īnd since the above is true for all regions, we have our familiar continuity-type equationĬonsider an infinite parallel-plate capacitor, with the lower plate (at ) carrying surface charge density, and the upper plate (at ) carrying charge density. Lucky for us, the magnetic field between the plates is (a) Determine all elements of the stress tensor, in the region between the plates. The terms on the diagonal will be zero, and So we can already tell that the off-diagonal terms will be zero, since they all contain two factors of, one of which will be zero. (c) What is the electromagnetic momentum per unit area, per unit time, crossing the xy plane (or any other plane parallel to that one, between the plates)? (b) Use at the boundary to determine the electromagnetic force per unit area on the top plate.
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