Electric and magnetic field energy

In summary: So, to make a long story short, it appears that the B^2 term comes from a form of the Lorentz force law. In terms of potentials, it arises when you take the curl of the vector potential, and it is a manifestation of the fact that the magnetic force does no work.In summary, the B^2 term in the general energy equation for electromagnetic fields comes from a form of the Lorentz force law and is a result of the fact that the magnetic force does no work. It can be derived by calculating the work required to increase the current in a solenoid or by considering the conservation of energy and the Poynting vector.
  • #1
Peeter
305
3
For electrostatics one can work out the energy of a charge configuration in terms of its associated field:

[tex]
U = \epsilon_0 \vec{E}^2/2
[/tex]

I've seen that the general energy of the field is given by:

[tex]
U = \epsilon_0 (\vec{E}^2 + c^2\vec{B}^2)/2
[/tex]

but am unsure how to show this. I suspect that a relativistic argument would be one way.

Another way, which doesn't work (or I did it wrong), is a calculation of the Hamiltonian for the (non-proper time formulation) of the Lorentz force Lagrangian. I get an energy that only includes the q\phi part as in electrostatics. This actually makes some sense since doing a line integral against a perpendicular field won't contribute.

I'd be interested to know two things:

1) A high level non-math description of where the B^2 term comes from.

2) Some hints for the math side of the question. What is a way (or some ways) that the E^2 + B^2 form of the field energy can be derived?
 
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  • #2
Peeter said:
1) A high level non-math description of where the B^2 term comes from.

2) Some hints for the math side of the question. What is a way (or some ways) that the E^2 + B^2 form of the field energy can be derived?

Remember the relationship between energy and force. Force is the spatial gradient of potential energy.

If you have the two fields E and B which do not vary over time, then

[tex]F = \triangledown U = \triangledown \epsilon_0(E^2 + c^2 B^2)/2 = \epsilon_0(2 E + 2 c^2 B)/2 = \epsilon_0 (E^2 + c^2 B)[/tex]

which is very similar to the form of the equation for the Lorentz force. The only part missing is the cross product of the charge's velocity, and I'm not sure exactly where that comes in. The equation you provided is missing any reference to the particle's velocity which (along with the charge) is necessary to go from magnetic potential to energy.
 
  • #3
Tac-Tics said:
Force is the spatial gradient of potential energy.

Not for this velocity dependent potential. The closest you'll probably get to a potential in this case is the via the Lagrangian

[tex]
L = T - V = mv^2/2 -q\phi + q A \cdot v
[/tex]

application of the Euler Lagrange equations will then give you the Lorentz force. Paraphrasing the Euler Lagrange equations in english, I'd say a more accurate description would be "the spatial gradient of the Lagrangian (Generalized Force) equals the time derivative of the velocity gradient of the Lagrangian).
 
  • #4
Peeter said:
I suspect that a relativistic argument would be one way.
No. People have tried to do that but all have failed.
Peeter said:
I'd be interested to know two things:
1) A high level non-math description of where the B^2 term comes from.
2) Some hints for the math side of the question. What is a way (or some ways) that the E^2 + B^2 form of the field energy can be derived?
1) Math is necessaryl at any level.
2) It is derived in most EM textbooks, even low level ones, usually starting with
[tex]dU/dt(matter)=\int{\bf j\cdot E}[/tex].
 
  • #5
One common way to derive the electric field energy density is to calculate the work required to increase the charge on a parallel-plate capacitor from 0 to +Q and -Q on the two plates, and relate it to the final electric field inside the capacitor and the volume of the capacitor. This gives you the equation that you posted:

[tex]u_E = \frac{1}{2} \epsilon_0 E^2[/tex]

Similarly, you can derive the magnetic field energy density by calculating the work required to increase the current in a solenoid from 0 to I, and relate it to the final magnetic field inside the solenoid and the volume of the solenoid. This gives you

[tex]u_B = \frac{1}{2} \frac{B^2}{\mu_0}[/tex]

The total electromagnetic field energy density is the sum of these two. You can then eliminate either [itex]\epsilon_0[/itex] or [itex]\mu_0[/itex] using the relationship

[tex]c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}[/tex]

which can be derived in connection with electromagnetic waves.
 
  • #6
clem said:
2) It is derived in most EM textbooks, even low level ones, usually starting with
.

I didn't see it in my old purcell text, but did a derivation of something similar to what you have above, where I got (I believe)

[tex]\partial U/\partial t + \nabla \cdot \bf P/c= -\bf j\cdot E}[/tex]

where P was the Poynting vector. In that case I started with U as defined in my initial post, and derived the conservation equation as a consequence of applying Maxwell's equations.

Now, by analogy, I see that the left hand side logically has the structure of a four vector energy momentum "divergence" so one could identify that with energy, but that's a whole lot different than identifying it as energy = force times distance as in the electrostatics case.
 
  • #7
jtbell said:
Similarly, you can derive the magnetic field energy density by calculating the work required to increase the current in a solenoid from 0 to I, and relate it to the final magnetic field inside the solenoid and the volume of the solenoid. This gives you

[tex]u_B = \frac{1}{2} \frac{B^2}{\mu_0}[/tex]

Thanks. I'll ponder that and give it a try.

jtbell said:
The total electromagnetic field energy density is the sum of these two.

It's not clear to me why this would have to be the direct sum of the two. For example, without some other guiding principle, why is it necessary that there is no E,B cross terms?
 
  • #8
I think I understand it now.

The - E . j term is a power density. For me I was able to obtain the energy/Poynting conservation relation by a purely mathematical manipulation of Maxwell's equations, but I didn't realize the significance of that last power density term. It can be related to work done by considering a line integral, moving a charge against the Lorentz force (and only the electric field ends up contributing to that integral)
 

Related to Electric and magnetic field energy

1. What is the difference between electric and magnetic field energy?

The main difference between electric and magnetic field energy is the type of force they exert. Electric fields exert forces on electric charges, while magnetic fields exert forces on moving electric charges. Additionally, electric fields are produced by stationary charges, while magnetic fields are produced by moving charges.

2. How is electric and magnetic field energy measured?

Electric field energy is measured in volts per meter (V/m), while magnetic field energy is measured in teslas (T). These units represent the strength of the field at a given point in space.

3. Can electric and magnetic fields be harmful to human health?

Electric and magnetic fields are a natural part of our environment and can be found in everyday objects such as cell phones, power lines, and household appliances. While high levels of exposure to these fields can be harmful, the levels found in our daily lives are generally considered safe.

4. How are electric and magnetic fields used in technology?

Electric and magnetic fields are used in various technologies, such as generators, transformers, motors, and MRI machines. They are also essential for the functioning of electronic devices, such as computers and cell phones.

5. Can electric and magnetic fields be shielded or blocked?

Yes, electric and magnetic fields can be shielded or blocked by certain materials. For example, metal objects can block magnetic fields, while insulators can block electric fields. However, it is important to note that complete shielding is difficult to achieve, and the strength of the fields can still be influenced by nearby sources.

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