CHAPTER 8 CONSERVATION LAWS

Transcription

1 CHAPTER 8 CONSERVATION LAWS Outlines 1. Charge and Energy 2. The Poynting s Theorem 3. Momentum 4. Angular Momentum 2 Conservation of charge and energy The net amount of charges in a volume V is given by, τ If the net charges in V is decreasing, then there must be some amount of charges are moving out of the boundary that enclose the V Combine equations (1) and (2), we have the continuity equation below. Poynting Vector We have shown in Chapter 2 that the work done to set up an electrostatic field is given by: (2.45) In Chapter 7, we showed that the work done to set up a static magnetic field can be written as: (7.34) Now if we take both and into account, it is natural to expect that the total work done to set up and as 3 4 Now we ll try to derive the previous equation. Assume that the work done on a charge by the fields is Lorentz force does not do any work,. From Ampere s law Substitute into equation (3) 5 From product rule #6 (6) This is Poynting theorem. 6 1

2 Let s define Energy density Energy flux The energy flux density is defined as the energy passing through an unit area per unit time. 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 The work done on the charges will increase their mechanical energy And let be the energy density of the fields We can re-write the Poynting theorem as This is continuity equation for the energy. This can also be viewed as the energy conservation law. 8 Basic concepts of Tensor In this chapter and chapter 12, tensor notation will be used. Here we will introduce some basic concepts of tensor. Tensor is an abstract mathematical object, whose properties is independent of the reference frames. Tensor describes the linear relationship between vectors scalars and other tensors. Now let s find out how various entities transform when we use different reference frames. Scalar (zero rank tensor) Scalar quantity is independent of coordinates. Vector (1 st rank tensor) Scalar Vector Tensor T Pv=nRT 9 If we let,, and 10 Here (x, y, z) and (x, y, z ) are the same vector in unprime and prime coordinates, and are the elements of the transformation matrix. Example: A rotation of angle θ about z-axis Tensor (2 nd rank tensor) We can see that 2 nd rank tensor has two indices, i and j, the indices run through the dimension of the space. In 3- D space, the 2 nd rank tensor has nine components. 11 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 contravariant indices such as. 12 2

3 Example of a tensor: Moment of inertia tensor. Momentum Now we define the moment of inertia tensor: Before we start to examining the stress tensor of the field, we briefly examine the filed patterns for moving charges. d = 13 Electric field of moving charge. Angular dependence of the radial component of the electric field of a point charge moving at different speed. 14 Magnetic field produced by a moving charge. Newton s 3 rd law in electrodynamics We can see that the on q 1 and q 2 satisfies the third law. But the magnetic force does not. cancel out. So the Newton s third law does not seem to be valid for these two moving charge particles. Breakdown of the Newton s third law is a very serious problem here, because momentum conservation law depends solely on the third law to ensure that ALL internal forces cancel each other out. Careful examination of the situation, leads us to realize that the electric field and the magnetic field not only carry energy, the fields also carry momentum. If we take into account the momentum of the particles and the fields, then we still have a general form of the law of conservation of energy and the law of conservation of Momentum Maxwell Stress Tensor The last term in eq. (7) can be re-written as Maxwell Stress Tensors are used to describe the conservation laws and force equation in electrodynamics. We start with a charge q moving in and field, the force experienced is Let be the force density due to the fields, and express charge density and current density in terms of fields. eq. (7) Substitute into eq. (7), we obtain (8)

4 Next we use the vector product rule (ii) on page 21, and let and, Maxwell stress tensor We obtain Re-arrange Similarly Substitute into eq. (8) on page 18, we end up with (10) The indices i and j are referred to the coordinates x, y, z, or just 1, 2, 3 as,, and. For the time being we just treat tensor as a matrix and as a matrix element. Maxwell stress tensor was developed to simplify the formulism Remember that dot product of a matrix with a vector is still a vector, and dot product of a vector with a matrix is also a vector. This can be applied to vector operator such as, Let j = y, Combine them together, we can see that (12) Compare eq. (12) with eq. (9) on page 19, we can see that it is the same as the first six terms of equation (9), EXAMPLE 8.2 Using the Poynting vector notation, we can re-write the above equation as In integral form, the total force on all charges in V is 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) Example

5 We will use eq. (8-22) to solve this problem. First of all we have to decide what volume to use. Clearly 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. Bowl In Cartesian coordinates The Maxwell stress tensor is given by Next we can see that the 2 nd 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. 25 There is no magnetic field, so we don t have to worry about the 2 nd term on the right-hand side. 26 From symmetry argument, we can see that the net force on the bowl is in the z-direction, so we only have to calculate Now we will try to calculate the z-component force on the bowl. At this point, we ignore the common constant factor, and only concentrate on the angular dependence parts The angular part is So the net force on the Bowl is For the Disk 29 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. 30 5

6 Here we ll repeat the example 8.2, but use the volume of the semiinfinite space of. (An infinite hemisphere) Bowl, because Disk (Infinite plane at z=0.) 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 Conservation of Momentum From Newton s second law, we can write the eq. (15) as Define the momentum in the electromagnetic field as We can write eq. (15) as where is the momentum density of the EM field. 33 We can see that Maxwell stress tensor is force per unit area is the momentum density of EM fields If we write eq. (17) differently, is a momentum density flux. From now on, we will use a new notation for the momentum density of EM fields, namely 34 If mechanical momentum is a constant, Therefore This is the continuity equation for momentum stored in EM field, and playing the role of momentum density flux. The continuity equation for energy is given by where and 35 PROBLEM 8.6 A charged parallel plate capacitor is placed in a uniform magnetic field as shown. a. Find the EM momentum between the plates. b. Let the capacitor slowly discharge through a resistive wire along the z-direction. Find impulse. c. If we gradually turn off the magnetic field. Find the impulse. 6

7 (a) The momentum density stored in the electromagnetic field is given by (c) If we turn off the magnetic field instead, the change of the magnetic flux will induce an emf (b) The impulse induced is given by l The electric field inside a parallel plate capacitor is / Where lis the area in the yz plane. l l l The net electric force on the charges on the plates is The impulse is given by EXAMPLE 8.3 =σabd= A long coaxial cable, of length l, consists of an inner conductor and an outer conductor with a radii of a & b. The inner conductor carries a uniform charges per unit length, and a current I to the right, the outer conductor has the opposite charge and current. What is the electromagnetic momentum stored in the field? Next we will look at example The fields between the two cylinders are The momentum in the E&M fields is given by, The Poynting vector is the energy flux The power transported is given by l l Where is this momentum coming from? Why the coaxial cable is not moving? In this case, there is hidden mechanical momentum associated with the flow of current, and it exactly cancel out with the momentum in the fields. We will come back to this in Chapter The situation is similar to the Problem 8.6 when a charged parallel plate capacitor is placed in a magnetic field. 42 7

8 Suppose we increase the resistance between the inner and outer conductors, the current decreases and the magnetic field also decreases, from Faraday s law, electric field will be generated in the z-direction such that This field will exert a force on both charges on inner and outer conductors l l The total momentum imparted to the cable is l This is exactly the same as the momentum stored in the EM fields. 43 Angular momentum We define the angular momentum density of the electromagnetic field as l Even static field can carry momentum or angular momentum as long as. Example Example 8.4 A long solenoid with radius R, n turns per unit length, and current I. Coaxial with the solenoid are two long cylindrical shells of length l, one inside carries +Q charge, and one outside carries Q charge as shown. When the current is gradually reduced, the cylinders begin to rotate. Where does the angular momentum come from? field between the cylinders is given by (a < s < b) Magnetic field inside the solenoid (s < R) The momentum density is The angular momentum density is given by l Total angular momentum is Summary When the current is turn off, the changing magnetic field induces circumferential electric field: Energy conservation law (s > R) (s < R) So the torque on the outer cylinder is (use the 1 st eq.) Similarly, the torque on inner cylinder is ----energy flux density / ---- energy density Momentum conservation law We can see that all works out momentum flux density momentum density 48 8