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Outlines ONSERVATION AWS 2/3/2016 CHAPTER 8 Outlines ONSERVATION AWS C L 1/28/2015 1. Charge and Energy Chapter 8 8 Chapter Momentum 2. The Poynting’s Theorem 3. Momentum 4. Angular Momentum 2 Conservation of charge and energy Poynting Vector We have shown in Chapter 2 that the work done to set up The net amount of charges in a volume V is given by an electrostatic field is given by: 1/28/2015 1/28/2015 ,·τ (2.45) If the net charges in V is decreasing, then there must be Chapter 8 Momentum Chapter 8 Momentum some amount of charges are moving out of the boundary In Chapter 7, we showed that the work done to set up a static that enclose the V magnetic field can be written as: · · · (7.34) Now if we take both and into account, it is natural to Combine equations (1) and (2), we have the continuity expect that the total work done to set up and as equation below. · 3 4 Now we’ll try to derive the previous equation. From product rule #6 Assume that the work done on a charge by the fields is · · · · · 1/28/2015 1/28/2015 Lorentz force does not do any work, ·. · · · · Chapter 8 Momentum Chapter 8 Momentum · · · · · From Ampere’s law · Substitute into equation (3) ∮ · (6) · · · 5 6 This is Poynting theorem. 1 2/3/2016 Let’s define The work done on the charges will increase their mechanical energy Energy density 1/28/2015 1/28/2015 Energy flux 8 Chapter Momentum And let be the energy density of the fields 8 Chapter Momentum The energy flux density is defined as the energy passing through an unit area per unit time. · We can re-write the Poynting theorem as · So the work done on the charges by the electromagnetic field is equal to the decrease in energy stored in the field, minus the energy flow out of the surface (boundary). 7 This is continuity equation for the energy. This can 8 also be viewed as the energy conservation law. Basic concepts of Tensor Now let’s find out how various entities transform when we use different reference frames. In this chapter and chapter 12, tensor notation will be used. Here we will introduce some basic concepts of 1/28/2015 1/28/2015 tensor. Scalar (zero rank tensor) Tensor is an abstract mathematical object, whose Chapter 8 Momentum Chapter 8 Momentum properties is independent of the reference frames. Scalar quantity is independent of coordinates. Tensor describes the linear relationship between vectors scalars and other tensors. Vector (1st rank tensor) Scalar Vector Tensor T Pv=nRT If we let , , and 9 10 Tensor (2 nd rank tensor) 1/28/2015 1/28/2015 Here (x, y, z) and (x’, y’, z’) are the same “vector” in un- We can see that 2nd rank tensor has two indices, i and j, prime and prime coordinates, and are the elements of Chapter 8 Momentum the indices run through the dimension of the space. In 3- Chapter 8 Momentum the transformation matrix. D space, the 2nd rank tensor has nine components. Example: A rotation of angle θ about z-axis The notation of tensor is similar to that of a matrix, but it can be more complicated. For example, tensor can have “covariant” indices such as or it can have 11 “contravariant” indices such as . 12 2 2/3/2016 Example of a tensor: Moment of inertia tensor. Momentum Before we start to examining the stress tensor of the field, 1/28/2015 we briefly examine the filed patterns for moving charges. 1/28/2015 · Chapter 8 8 Chapter Momentum 8 Chapter Momentum Now we define the moment of inertia tensor: d Angular dependence of the radial = · Electric field of moving charge. component of the electric field of a point charge moving at different speed. · 13 14 Breakdown of the Newton’s third law is a very serious problem here, because momentum conservation law 1/28/2015 depends solely on the third law to ensure that ALL 1/28/2015 internal forces cancel each other out. Magnetic field produced by a moving charge. Chapter 8 Momentum Chapter 8 Momentum Careful examination of the situation, leads us to realize rd Newton’s 3 law in electrodynamics that the electric field and the magnetic field not only carry energy, the fields also carry momentum. We can see that the on q1 and q2 satisfies the third law. But the magnetic force does If we take into account the momentum of the particles .not cancel out. So the and the fields, then we still have a general form of the Newton’s third law does not law of conservation of energy and the law of seem to be valid for these conservation of Momentum. two moving charge particles. 15 16 Maxwell Stress Tensor The last term in eq. (7) can be re-written as Maxwell Stress Tensors are used to describe the 1/28/2015 1/28/2015 conservation laws and force equation in electrodynamics. We start with a charge q moving in and field, the Chapter 8 Momentum Chapter 8 Momentum force experienced is · Substitute into eq. (7), we obtain Let be the force density due to the fields, and express · charge density and current density in terms of fields. (8) · eq. (7) 17 18 · 3 2/3/2016 Next we use the vector product rule (ii) on page 21, Maxwell stress tensor and let and , 1/28/2015 1/28/2015 We obtain · (10) · Re-arrange Chapter 8 Momentum The indices i and j are referred to the coordinates x, y, z, or 8 Chapter Momentum Similarly · just 1, 2, 3 as , , and . For the time being we just treat tensor as a matrix and as a matrix Substitute into eq. (8) on page 18, we end up with element. · · · · 19 20 Maxwell stress tensor was developed to simplify the formulism. Remember that dot product of a matrix with a vector is Let j = y, still a vector, and dot product of a vector with a matrix is also a vector. 1/28/2015 1/28/2015 · ≡ Chapter 8 Momentum Chapter 8 Momentum · ≡ Combine them together, we can see that This can be applied to vector operator such as , · · · · · · (12) 21 22 Compare eq. (12) with eq. (9) on page 19, we can see that EXAMPLE 8.2 it is the same as the first six terms of equation (9), 1/28/2015 Determine the net force on the “northern” hemisphere · of a uniformly charged solid sphere of radius R and charge Q. (Same as in Problem 2.47) Using the Poynting vector notation, we can re-write the Chapter 8 Momentum above equation as · In integral form, the total force on all charges in V is · Example 8.2 23 4 2/3/2016 We will use eq. (8-22) to solve this problem. Bowl 1/28/2015 1/28/2015 · Chapter 8 8 Chapter Momentum 8 Chapter Momentum First of all we have to decide what volume to use. Clearly In Cartesian coordinates the simplest way is to use the volume of the hemisphere itself. The surface integral part can be divided into the “Bowl” and the “Disk” as shown on the drawing. The Maxwell stress tensor is given by Next we can see that the 2nd term on the right is equal to zero, because there is no magnetic field and also because it is a statics problem and no time dependence. There is no magnetic field, so we don’t have to worry 25 about the 2nd term on the right-hand side. 26 From symmetry argument, we can see that the “net force” on the Now we will try to calculate the z-component force on the “bowl”. At “bowl” is in the z-direction, so we only have to calculate this point, we ignore the common constant factor, and only concentrate on the angular dependence parts 1/28/2015 1/28/2015 · · · . Chapter 8 Momentum Chapter 8 Momentum The angular part is 27 28 So the net force on the Bowl is · 1/28/2015 1/28/2015 · · · For the Disk Chapter 8 Momentum Chapter 8 Momentum Please note that when we use eq. (1), it does not matter what volume we used. This situation is very similar to using “Gauss Law”, you can choose any Gaussian surface. · · · 29 30 5 2/3/2016 Here we’ll repeat the example 8.2, but use the volume of the semi- infinite space of →∞. (An infinite hemisphere) Bowl · 1/28/2015 1/28/2015 , because Disk (Infinite plane at z=0.) Chapter 8 8 Chapter Momentum 8 Chapter Momentum We only need to calculate from r=R to r=∞. This is exactly the same as we calculated before for the force on the “bowl”. 31 32 Conservation of Momentum We can see that From Newton’s second law, we can write the eq. (15) as 1/28/2015 1/28/2015 Maxwell stress tensor is force per unit area · · Chapter 8 Momentum is the momentum density of EM fields Chapter 8 Momentum Define the momentum in the electromagnetic field as If we write eq. (17) differently, · · ℘ ℘ We can write eq. (15) as is a momentum density flux. ℘ ℘ · From now on, we will use a new notation for the momentum density of EM fields, namely 33 34 where ℘ is the momentum density of the EM field. ℘ If mechanical momentum is a constant, PROBLEM 8.6 A charged parallel plate capacitor is placed in a uniform magnetic field as shown.
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