Maxwell equation involving the magnetic field

This is the Maxwell equation we were looking for.In summary, the conversation discusses the thermodynamic identity for free energy in the presence of an internal magnetic field, and how to derive the appropriate Maxwell equation to show that the magnetic moment (M) is constant at a given temperature (T) when the internal magnetic field (B) is kept constant. The equation for this is dM/dT = 0. The conversation also mentions the attempt to isolate M in the dF equation, but getting stuck and finding SdT to be zero. However, by taking the derivative of the thermodynamic identity for free energy with respect to T and rearranging, the desired Maxwell equation is derived.
  • #1
ken~flo
6
0
1. In the presence of an internal magnetic field B, the thermodynamic identity for the free energy F is (assuming V is kept constant)
dF = -SdT - MdB
where M is the total magnetic moment (magnetization times volume) of the system. Derive an appropriate Maxwell equation to show that
(dM/dT) = 0
at constant B, T = 0.


2. M = -(dF/dT) at constant T


3. I tried isolating M in the dF equation, then taking the derivative with respect to T of the whole thing, which is where I got stuck. I also found SdT to be zero, which may or may not be applicable here? Any help is much appreciated.
 
Physics news on Phys.org
  • #2
4. The Maxwell equation you are looking for is:dM/dT = 0 This equation states that the magnetic moment (M) is constant at a given temperature (T), which is what we would expect for a system with a constant internal magnetic field (B). To derive this equation, simply take the derivative of the thermodynamic identity for the free energy with respect to T, while keeping B constant:dF/dT = -S - MdB/dT = -MdB/dTRearranging this equation, we get:dM/dT = -(dF/dT)/dB = 0 This equation states that the rate of change of magnetic moment (M) with respect to temperature (T) is zero when the magnetic field (B) is held constant.
 

What is the Maxwell equation involving the magnetic field?

The Maxwell equation involving the magnetic field is the fourth of the four Maxwell equations, also known as Maxwell's equations of electromagnetism. It describes the relationship between the magnetic field and the electric current in a given region of space.

What is the mathematical representation of the Maxwell equation involving the magnetic field?

The mathematical representation of the Maxwell equation involving the magnetic field is:
∇ x B = μ0J + ε0μ0∂E/∂t
Where ∇ represents the gradient operator, B represents the magnetic field, μ0 is the permeability of free space, J represents the electric current, E represents the electric field, ε0 is the permittivity of free space, and ∂/∂t represents the time derivative.

What is the significance of the Maxwell equation involving the magnetic field?

The Maxwell equation involving the magnetic field is significant because it is one of the fundamental laws of electromagnetism, describing the relationship between electricity and magnetism. It also allows us to understand and predict the behavior of electromagnetic waves, which are essential for modern technologies such as radio, television, and wireless communication.

How is the Maxwell equation involving the magnetic field related to the other Maxwell equations?

The Maxwell equation involving the magnetic field is related to the other three Maxwell equations through the principle of electromagnetic duality. This principle states that the electric and magnetic fields are two different manifestations of the same fundamental force, and can be converted into each other by a change in reference frame. Therefore, the equations describing these fields are interconnected and must be considered together to fully understand the behavior of electromagnetic phenomena.

What are some applications of the Maxwell equation involving the magnetic field?

The Maxwell equation involving the magnetic field has various applications, including the design and operation of electric motors, generators, and transformers. It is also essential for understanding the behavior of magnetic materials and the magnetic fields of celestial bodies. Additionally, it is used in the development of technologies such as magnetic levitation and magnetic resonance imaging (MRI).

Similar threads

  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Electromagnetism
Replies
5
Views
155
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
3K
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
971
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Back
Top