(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. 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. Lucky for us, the magnetic field between the plates is (a) Determine all elements of the stress tensor, in the region between the plates. 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. 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. Which is the momentum density within the fields. 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. The first term on the right is related to the momentum stored in the electromagnetic field. 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. 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). 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. 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. Suppose we have a volume containing some distribution of charge, current, and electromagnetic fields. In the interest of brevity, we'll skip around a bit and leave the full derivations for the real textbook. 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. Starting with the basic Coulomb/Lorentz laws, we'll write down an expression for the electromagnetic force on charges in a volume. The way to recover conservation of momentum proceeds the same way we recovered the conservation of energy via the Poynting vector. 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. 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. Conservation of electromagnetic momentumĪs it turns out, if you disregard the momentum associated with electromagnetic fields, Newton's laws appear not to work out! Consider a basic system in cartesian coordinates of two moving point charges:.As friction increases, momentum decreases.2.2 - Divergence and Curl of Electrostatic FieldsĨ.2.3 Conservation of Electromagnetic Momentum Does friction affect the conservation of momentum?Īnswer: Yes, friction affects momentum. \(P_\) are the final velocities of the bodies. On the off chance, if i and f indicate the initial and final momentum of objects in a framework, then the principle of conservation of momentum says that Practically this means we need to incorporate the two objects and anything that applies a force to any of the objects for any timeframe in the system. For this situation, an isolated system is one that is not acted on by any external force to the system – i.e., there is no external impulse. Likewise, with the other conservation standards, there is a catch: conservation of momentum applies just to an object of an isolated system. 2 FAQs on Conservation of Momentum Introduction to Conservation of Momentum
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