What are the operators here and how are these formulas derived?

In summary, the operators in question refer to mathematical symbols or functions that manipulate variables in formulas. These formulas are derived through a combination of foundational mathematical principles, such as algebraic rules, calculus, and logical reasoning, which help establish relationships between different variables and enable problem-solving in various contexts.
  • #1
Ren_Hoek
11
0
1719990100132.png
1719990106169.png


In (23), are grad and div some kind of scalar operators comparing to and ?
because tbh I dont know how turns into even given .

1719990166994.png

tbh I don't really get it. First, Is \sigma, which I assume is electrical conductivity, related to static magnetic field?
 
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  • #2
No idea about this particular situation, but typically div stands for divergence and grad for gradient, these have well defined meanings.
 
  • #3
Ren_Hoek said:
In (23), are grad and div some kind of scalar operators comparing to $\nabla$ and $\nabla\times$?
because tbh I dont know how $\text{curl}(\mu^{-1}\text{curl}\textbf{A})$ turns into $\text{div}\mu^{-1}\text{grad}A_z$ even given $A=(0,0,A_z)^T$.
tbh I don't really get it. First, Is \sigma, which I assume is electrical conductivity, related to static magnetic field?
Hint: On Physics Forums, "single dollar signs" don't denote LaTex. Instead, inline formulas are sandwiched between "double pound signs" and display formulas between "double dollar signs". (See the LaTeX Guide at the bottom left of the posting screen.)

Below I've reformatted your text to make the LaTeX readable:

In (23), are grad and div some kind of scalar operators comparing to and ?​
because tbh I dont know how turns into even given .​
tbh I don't really get it. First, Is , which I assume is electrical conductivity, related to static magnetic field?​
And to answer at least your first question: yes, , , and .
 
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  • #4
renormalize said:
Hint: On Physics Forums, "single dollar signs" don't denote LaTex. Instead, inline formulas are sandwiched between "double pound signs" and display formulas between "double dollar signs". (See the LaTeX Guide at the bottom left of the posting screen.)

Below I've reformatted your text to make the LaTeX readable:

In (23), are grad and div some kind of scalar operators comparing to and ?​
because tbh I dont know how turns into even given .​
tbh I don't really get it. First, Is , which I assume is electrical conductivity, related to static magnetic field?​
And to answer at least your first question: yes, , , and .
thank you.
No matter how I calculate (calculating directly or using ), it is



for me so I don't know what this section of the book is talking about.
 
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  • #5
I think grad before seems redundant here, as here, so as long as it is much more understandable, but I couldn’t find evidence of it and maybe it's related to strong physics meaning which I'm not good at.
 
  • #6
Ren_Hoek said:
I think grad before seems redundant here, as here, so as long as it is much more understandable, but I couldn’t find evidence of it and maybe it's related to strong physics meaning which I'm not good at.
Don't forget that by eq.(28) in your posted text, the permeability can also be a function of the transverse-position (as well as time): . Just use the basic definition of the curl in cartesian coordinates and "turn the crank" twice to get the result in eq.(22) of your text:No "strong physics" involved!
 
  • #7
renormalize said:
Don't forget that by eq.(28) in your posted text, the permeability can also be a function of the transverse-position (as well as time): . Just use the basic definition of the curl in cartesian coordinates and "turn the crank" twice to get the result in eq.(22) of your text:No "strong physics" involved!
Thank you, I see.
But how does turn into (0, 0) or are they simply neglected since we're considering the z axis? or was my calculation wrong?
 
  • #8
I think as suggested in (28), it is also suggested in the expression as it does not contain , and will make the second order derivative to 0 in the above case.
But I still don't fully understand the physical meaning of it.
 
  • #9
Ren_Hoek said:
But I still don't fully understand the physical meaning of it.
I can't comment on "physical meaning" since the partial text you posted doesn't describe the configuration that is being analyzed. Maybe it's something like a long, current-carrying solenoid coil or a waveguide? A configuration, relatively long in the - (longitudinal) direction and relatively short in the -,- (transverse) directions, can sometimes be well-approximated physically, at least away from the ends, by postulating that the currents and fields have a specific functional form . Examples are a DC solenoid () or an electromagnetic waveguide propagating an oscillating field ().
 
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  • #10
renormalize said:
I can't comment on "physical meaning" since the partial text you posted doesn't describe the configuration that is being analyzed. Maybe it's something like a long, current-carrying solenoid coil or a waveguide? A configuration, relatively long in the - (longitudinal) direction and relatively short in the -,- (transverse) directions, can sometimes be well-approximated physically, at least away from the ends, by postulating that the currents and fields have a specific functional form . Examples are a DC solenoid () or an electromagnetic waveguide propagating an oscillating field ().
Thank you.
1720028111679.png

I think it simulates this coil model.
 
  • #11
Ren_Hoek said:
I think it simulates this coil model.
OK, that clarifies your configuration. From Google, there are a fair number of references about modeling C-electromagnets. For example, CERN Dirac C-magnet:
C-magnet config.png

Since the transverse size ( directions) is large compared to the -extent, your 2D FEM model is only an approximate representation of the actual magnet. As the reference says:
Modeling.png

Here's the full 3D model of the magnetic field:
C-magnet 3D model.png

Of course, in the analysis and model you've cited, you apparently have the additional complication of time-dependent currents and fields. So it's a fairly complex problem.
 
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  • #12
renormalize said:
OK, that clarifies your configuration. From Google, there are a fair number of references about modeling C-electromagnets. For example, CERN Dirac C-magnet:
View attachment 347767
Since the transverse size ( directions) is large compared to the -extent, your 2D FEM model is only an approximate representation of the actual magnet. As the reference says:
View attachment 347768
Here's the full 3D model of the magnetic field:
View attachment 347769
Of course, in the analysis and model you've cited, you apparently have the additional complication of time-dependent currents and fields. So it's a fairly complex problem.
Thank you, I will take a further look into this model.
 

FAQ: What are the operators here and how are these formulas derived?

What are operators in the context of mathematics and physics?

Operators are mathematical entities that act on elements of a space to produce other elements of that space. In mathematics, they can represent functions, transformations, or operations like differentiation and integration. In physics, operators often represent observable quantities, such as momentum or energy, and are used in quantum mechanics to describe the state of a system.

How are mathematical operators defined?

Mathematical operators are defined by their action on functions or vectors. For example, the differential operator (d/dx) acts on a function by providing its derivative, while the integral operator (∫) sums up values over a specified interval. The precise definition depends on the context, such as linear operators in linear algebra or differential operators in calculus.

What are some common examples of operators?

Common examples of operators include the differential operator (∂/∂x), which is used to calculate derivatives, the Laplacian operator (∇²), which is used in physics to describe diffusion processes, and matrix operators in linear algebra that transform vectors in a vector space. Each of these operators has specific mathematical properties and applications.

How are formulas involving operators derived?

Formulas involving operators are derived using mathematical principles such as calculus, linear algebra, and functional analysis. For instance, the derivation of the formula for the action of the differential operator on a function involves applying the definition of the derivative. In quantum mechanics, the derivation of the Schrödinger equation involves applying operators to wave functions and using principles of wave-particle duality.

What is the significance of operators in quantum mechanics?

In quantum mechanics, operators are crucial as they represent physical observables, such as position and momentum. The state of a quantum system is described by a wave function, and operators act on this wave function to extract measurable quantities. The mathematical framework of quantum mechanics relies heavily on operator theory, particularly in the formulation of the uncertainty principle and the evolution of quantum states.

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