How do I calculate the derivative of the function T_el with respect to yd?

In summary, the person is trying to calculate the energy resolution of a magnetic spectrometer and is having difficulty with the derivation. They ask for help and provide a summary of the problem and possible solutions.
  • #1
1Keenan
101
4
Hello,

I would need some help in calculating the derivative of the function T_el in the attached image.
I want to calculate d T_el /d yd, where yd is the variable and it appears in the term I called A_elSide. Its expression is again in the image.
Numbers you see are not important.Just to explain what I am trying to do:
I want to calculate energy resolution of a magnetic spectrometer (which means distance between two energy point) and it can be calculated as (dT/dy)*spotsize, basically the derivative of the energy with respect to the position on the detector and multiplied by the "nominal" beam spot size.

This means I need to express the energy as a function of the beam position.
Beam position at the detector plane is yd=R*sin(theta)+(d*tg(theta)

First term (R*sin(theta)) is the y coordinate at the magnet output, second term ((d*tg(theta) ) is the additional displacement in the drift d between magnet and detector.
Kinetic energy is in the bending radius R, so doing some math I got an expression for the kinetic enexrgy, which is T_el in the pictureExpression for T_el has an element A_elSide, cotaining the position on the detector (yd), which is the variable with respect I should do the derivative.

An now I need help, I did it already 4 times and I got 4 different results...
Any help?
 

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  • #2
1Keenan said:
I got 4 different results...
I don't see any of them. Telepathic capabilities severely limited.
And please, ##\LaTeX##, not pictures. Do you want us to do it for you ?
 
  • #3
no, I don't want you to do it for me, off course!
I didn't posted them because they are wrong.

My strategy is to derive it as a sqrt((a+f(x))^2) but I guess it doesn't work properly...
 
  • #4
To me it seems the tough part is inverting
1Keenan said:
yd=R*sin(theta)+(d*tg(theta) )
so I wonder if ##\theta## is small enough to approximate linearly.
(by the way, the thing is called tan, not tg )

1Keenan said:
My strategy is to derive it as a sqrt((a+f(x))^2)
So what is $${d\over dz} \, \sqrt {a+z^2} \quad ? $$
1Keenan said:
I didn't posted them because they are wrong.
Post what you think is the best one and solicit comments
 
  • #5
Tan or tg is the same...
I'll redo the calculation tomorrow, I think I made a mess with all the other constant terms
 
  • #6
ok, I did calculation and the most reasonable result is attached here.

There is a problem, it doesn't work at all...
any help?
 

Attachments

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  • #7
I tried to help in post #4. Did it not help ?
 
  • #8
Yes, you did.
I forgot to tell you that theta is not small to approximate sin(theta) = theta.

I think I'm messing up with the constants. Or do you see any huge mistake in the derivative?
 

What is the function T_el?

The function T_el represents the temperature of an element, which can be calculated using various factors such as heat transfer, mass, and specific heat capacity.

What does it mean to calculate the derivative of a function?

Calculating the derivative of a function means finding the rate of change of the function at a specific point. In other words, it measures how much the output of the function changes when the input changes.

Why do we need to calculate the derivative of T_el with respect to yd?

Calculating the derivative of T_el with respect to yd allows us to understand how changes in the variable yd affect the temperature of the element. This can be useful in determining optimal values for yd to achieve a desired temperature.

What is the process for calculating the derivative of T_el with respect to yd?

The process for calculating the derivative of T_el with respect to yd involves using the rules of differentiation, such as the power rule and the chain rule, to find the derivative. It is important to also consider any constants or coefficients in the original function.

What are some real-world applications of calculating the derivative of T_el with respect to yd?

One possible application is in designing heating or cooling systems for buildings. By understanding how changes in yd affect the temperature of an element, engineers can optimize the design to achieve a desired temperature. The derivative of T_el with respect to yd can also be used in thermodynamics and heat transfer studies.

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