2/3/2016
CHAPTER 8 Outlines ONSERVATION AWS
C L 1/28/2015 1. Charge and Energy Chapter 8 Momentum Chapter 8 2. The Poynting’s Theorem
3. 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 Momentum 8 Chapter Momentum 8 Chapter 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 8 Chapter Momentum 8 Chapter · · · · · 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 Momentum Chapter 8 And let be the energy density of the fields Momentum Chapter 8 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 Momentum 8 Chapter Momentum 8 Chapter 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 8 Chapter the indices run through the dimension of the space. In 3- Momentum 8 Chapter 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 Momentum Chapter 8 Momentum Chapter 8 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 8 Chapter Momentum 8 Chapter 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 8 Chapter Momentum 8 Chapter 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 Momentum 8 Chapter The indices i and j are referred to the coordinates x, y, z, or Momentum Chapter 8 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 · ≡ Momentum 8 Chapter Momentum 8 Chapter · ≡ 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 Momentum 8 Chapter 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 Momentum Chapter 8 Momentum Chapter 8 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 8 Chapter Momentum 8 Chapter 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 8 Chapter Momentum 8 Chapter
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 Momentum Chapter 8 Momentum Chapter 8 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 8 Chapter is the momentum density of EM fields Momentum 8 Chapter 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. 1/28/2015 · · a. Find the EM momentum between the plates. b. Let the capacitor slowly discharge through a resistive wire along the z-direction. Find impulse. Chapter 8 Momentum 8 Chapter Therefore c. If we gradually turn off the magnetic field. Find the · impulse.
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 · 35 where and
6 2/3/2016
(a) The momentum density stored in the electromagnetic (c) If we turn off the magnetic field instead, the change of the field is given by magnetic flux will induce an emf 1/28/2015 1/28/2015 ℘ ·
(b) The impulse induced is given by Momentum Chapter 8 Momentum Chapter 8
Where ℓ is the area in the yz plane. ℓ ℓ ℓ ℓ The electric field inside a parallel plate capacitor is The net electric force on the charges on the plates is /
· ∆ 37 38
The impulse is given by EXAMPLE 8.3 A long coaxial cable, of length l, consists of an inner conductor
1/28/2015 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 Momentum 8 Chapter charge and current. What is the electromagnetic momentum =σABd= · stored in the field?
Next we will look at example 8.3 39
The fields between the two cylinders are The momentum in the E&M fields is given by
, 1/28/2015 ℓ 1/28/2015 ℓ
The Poynting vector is the energy flux Momentum 8 Chapter Where is this momentum coming from? Why the Momentum 8 Chapter 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 The power transported is given by Chapter 12.
The situation is similar to the Problem 8.6 when a charged parallel plate capacitor is placed in a magnetic · field. 41 42
7 2/3/2016
Suppose we increase the resistance between the inner and Angular momentum outer conductors, the current decreases and the magnetic field also decreases, from Faraday’s law, electric field will We define the angular momentum density of the be generated in the z-direction such that 1/28/2015 electromagnetic field as 1/28/2015
Momentum Chapter 8 Momentum Chapter 8 ℓ ℘ This field will exert a force on both charges on inner and outer conductors ℓ ℓ
The total momentum imparted to the cable is Even static field can carry momentum or angular ℓ momentum as long as . 43 Example 8.4 44 This is exactly the same as the momentum stored in the EM fields.
Example 8.4 field between the cylinders is given by (a < s < b) A long solenoid with radius R, n 1/28/2015 1/28/2015 turns per unit length, and current Magnetic field inside the solenoid I. Coaxial with the solenoid are Chapter 8 Momentum 8 Chapter (s < R) Momentum 8 Chapter two long cylindrical shells of length l, one inside carries +Q The momentum density is charge, and one outside carries –Q ℘ charge as shown. When the current is gradually reduced, the The angular momentum density is given by cylinders begin to rotate. Where does the angular momentum come ℓ ℘ from? Total angular momentum is 45 46
Summary When the current is turn off, the changing magnetic Energy conservation law field induces circumferential electric field: 1/28/2015 · 1/28/2015 (s > R)
----energy flux density Momentum 8 Chapter (s < R) st / ---- energy density So the torque on the outer cylinder is (use the 1 eq.) Chapter 8 Momentum Momentum conservation law Similarly, the torque on inner cylinder is · ℘ ℘ ------momentum flux density We can see that all works out 47 ℘ ------momentum density 48
8