Questions tagged [approximations]
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912 questions
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Useful approximation for net field vector near a pair of cubic magnets
Aim
I want to design a position sensor based on the net field surrounding a pair of magnets as illustrated below. The setup is intended to detect angular displacement on an axis perpendicular to the ...
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Trying to find wave equation for a string by abusing notation (algebraically manipulating differentials)
I'm trying to derive the wave equation for a string of mass density $\mu$ with an equal tension $T$ being applied at both ends at an angle $\theta_x$ relative to the $x$-axis. I've taken an ...
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Asymptotic integral in Peskin & Schroeder, Problem 6.3
The question is about P&S QFT Problem 6.3. In question (a), the contribution to $a = \frac{g - 2}{2}$ from Higgs boson is calculated, the result is:
$$
\delta a = \frac{\lambda^2}{16 \pi^2} \int_0^...
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Help understand Stirling's formula for large $n$
I have just started reading Concepts in Thermal Physics by Blundell & Blundell. On page 440 in the appendix two Stirling's formulas are derived as approximations for $\ln n!$, namely,
$$\ln n! \...
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Problem with the Sommerfeld Expansion for the Pressure Integral at Finite Temperature
i want to approximate my pressure integral by using the Sommerfeld expansion.
My pressure integral has the form:
\begin{align}
P = \int_0^{\infty} c \cdot \frac{k^4}{E(k)} \cdot \frac{1}{\exp\left(\...
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How do you derive the condition for which the quasi-magnetostatic approximation holds?
Premise:
I am talking about the 'Dynamic and Quasi-static fields' Chapter 14 from Zangwill's Electrodynamics text. In it, he uses order of magnitude approximations to arrive at a condition relating ...
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Point charge in the 3D torus: is there an approximate result for its potential?
Assume to have a unit point charge in a flat 3D torus of side $L$. There is also a uniform neutralising background, so that the total charge is exactly zero. Finding the electric potential means ...
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How to obtain approximation for expectation value in this statistical mechanics problem? [closed]
We have a system at temperature $k_B T = \tau$, and it has $n+1$ possible states, each with energy $i\epsilon$, where $i$ ranges from $0$ to $n$. The expectation value of $i$ is thus
$$\langle i \...
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Why does the film method require an approximation but the force method doesn't in computing the capillary rise in this arrangement?
In the arrangement shown below, if the surface tension of the given liquid is $T$, its density is $\rho$, angle of contact $\theta=0^\circ$ and $R_2-R_1 \ll R_1$ and $R_2$, evaluate $h$.
$\textbf{...
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Closed form for the Ideal Gas Constant in 1D, 2D, and 3D
The value of the Ideal Gas Constant, $R$, is experimentally determined to be about $8.314... \frac{J}{K⋅mol}$.
It is used in the ideal gas law:
$$PV = nRT$$
But when I look at the ideal gas ...
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Approximation of spacial derivatives of wave field
I recently came across a question whereby it is required to show that the solutions to the following equations are equivalent for some condition on the wavelength, $\lambda$, of the wave field $u(x,t)$...
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Fermi's golden rule for total transition rate [duplicate]
Im following Mark Thomson's Modern Particle Physics section 2.3.6 but I have question in the proceedure followed from equation 2.46 to 2.48
Starting from equation 2.46:
$$d \Gamma_{fi} = \frac{1}{T} ...
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Quasi-static EM and derivatives a factors
For the quasi-static approximation for EM, Zangwill and Jackson talk about approximating derivatives. Here, I’m paraphrasing Zangwill as he provides more detail.
Assume charge density varies slowly. ...
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Ignoring higher order terms leading to same result as "exact" calculation
To illustrate the context, a thermodynamics problem:
"Thermodynamic properties of interfaces between two phases are described by the surface tension $S$. It is defined by the work required to ...
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When is wave optics no longer a good approximation?
In optics, we know that whenever we want to take into account border effects or, more mathematically, whenever the wavelength of the light is comparable to the size of our object, we can no longer use ...
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