Introduction to Electrodynamics/Chapter 7/5/4
7.5.4 Conservation of Momentum
Concepts
- Momentum is stored in the E and B vectors through the Poynting Vector S.
- <math>-\overset{\leftrightarrow}{\mathbf{T
}}
The introduction of Poynting's Vector and Maxwell's Stress Tensor now give us a framework where we can talk about the conservation of momentum.
Since the total force on a volume is (See the previous section:
- <math>\vec{F} = \oint_S \overset{\leftrightarrow}{\mathbf{T}}\cdot d\vec{a} - \epsilon_0 \mu_0 {d \over dt}\int_V \vec{S}\ d\tau\;</math>
And <math>\vec{F} = d/dt \vec{p}\;</math> (force is the change in momentum over time):
- <math>{d \vec{p} \over dt} = \oint_S \overset{\leftrightarrow}{\mathbf{T}}\cdot d\vec{a} - \epsilon_0 \mu_0 {d \over dt}\int_V \vec{S}\ d\tau\;</math>
This means that the field S is "absorbing" some of the momentum that otherwise would be transferred to the object in mechanical motion. Or, in other words, the electric and magnetic fields themselves are being acted upon, and experience a force, and absorb some of the momentum.
The momentum stored in the E and B fields is:
- <math>\vec{p}_{EB} = \mu_0 \epsilon_0 \int_V \vec{S}\ d\tau\;</math>
And the momentum per unit volume stored in the same:
- <math>\vec{\mathfrak{p}}_{EB} = \mu_0 \epsilon_0 \vec{S}\;</math>
If <math>\vec{\mathfrak{p}}\;</math> is the mechanical momentum per unit volume, and so, applying the Divergence Theorem to the surface integral, we can rewrite the momentums as:
- <math>-{\partial \over \partial t} (\vec{\mathfrak{p}} + \vec{\mathfrak{p}}_{EB})
= \nabla \cdot (-\overset{\leftrightarrow}{T})\;</math>
This means that <math>-\overset{\leftrightarrow}{T}\;</math> acts as momentum flux density. <math>\overset{\leftrightarrow}{T}\;</math> is the electromagnetic stress acting on a volume, while <math>-\overset{\leftrightarrow}{T}\;</math> is the flow of momentum through the fields themselves.
S is the energy per unit volume per unit time carried by the electromagnetic fields. <math>\mu_0 \epsilon_0 \vec{S}\;</math> is the momentum per unit volume stored in the fields.
We can no longer suppose that the energy from a charge or current configuration is stored in the charges and currents themselves. We are now forced to admit that the fields themselves store the energy, regardless of what the charges and currents are doing somewhere else.
We have resolved the contradiction in Newton's Third Law applied to electrodynamics by showing that charged particles and currents do not interact directly with each other, but transfer their energy and momentum through the fields themselves.
Now, you can no longer point to Coulomb's Law as the truth. It is a convenient crutch that applies only in static circumstances. The reality is this:
- Charged particles or changing magnetic fields establish an electric field.
- Currents or changing electric fields establish a magnetic field
- Moving charged particles put energy and momentum into, or take energy or momentum from, those fields.
Mastery
- Follow all the details of example 17. Fill in all the missing steps until you can solve it on your own.
- Do problems at the end of the section.
- Commit to memory the equations relating momentum, S, and T.
Testing Yourself
- How do force and momentum relate?
- Suppose a force is exerted on a region of space, perhaps occupied by mass or charge or currents. Must that force result in a change in velocity of that region in space?
- How does T represent force and momentum? Give precise terms.
- How does S represent energy and momentum? Give precise terms.
- Recall the equation relating mechanical momentum, electromagnetic momentum, and T.
- What configurations of E and B carry momentum? Give a rule for any E and B field configuration that does not carry momentum.
- Is it possible to establish an E field without transferring momentum to it? How would this be done?
Example 17
