Chapter 3: Problem 5
What kind of flux (flux of what?) is current density (current per area)?
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These are the key concepts you need to understand to accurately answer the question.
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Chapter 3: Problem 5
What kind of flux (flux of what?) is current density (current per area)?
These are the key concepts you need to understand to accurately answer the question.
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Evaluate these definite integrals. Hint: Use symmetry. (a) $$ \int_{-10}^{10} x^{3} e^{-x^{2}} d x $$ (b) $$ \int_{-\infty}^{\infty} \frac{x^{3}}{1+7 x^{2}+18 x^{8}} d x $$ and (c) $$ \int_{0}^{\infty} \frac{\ln x}{1+x^{2}} d x . $$
At each step in triplet solitaire (Problem \(3.1 .2\) ), there are three possible moves, depending on which pair of numbers from among \(a, b\), and \(c\) you choose to replace. Describe each of the three moves as a rotation in space. That is, for each move, give the rotation axis and the angle of rotation.
On the logarithmic scale in the text, the gap between 1 and 10 is approximately \(10.5\) centimeters. If the scale were extended to include numbers up to 1000 , how large would the gap between 10 and 1000 be?
Here is an inertia tensor (the generalization of moment of inertia) of a particular object, calculated in an ill-chosen (but Cartesian) coordinate system: $$ \left(\begin{array}{lll} 4 & 0 & 0 \\ 0 & 5 & 4 \\ 0 & 4 & 5 \end{array}\right) $$ a. Change the coordinate system to a set of principal axes, where the inertia tensor has the diagonal form $$ \left(\begin{array}{ccc} I_{x x} & 0 & 0 \\ 0 & I_{y y} & 0 \\ 0 & 0 & I_{z z} \end{array}\right) $$ and give the principal moments of inertia \(I_{\mathrm{xx}}, I_{\mathrm{yy}}\), and \(I_{\mathrm{zz}}\). Hint: Which properties of a matrix are invariant when changing coordinate systems? b. Give an example of an object with a similar inertia tensor. Rhetorical question: In which coordinate system is it easier to think of such an object? This problem was inspired by a problem on the physics written qualifying exam during my days as a PhD student. The problem required diagonalizing an inertia tensor, and there was too little time to rederive or even apply the change-of- basis formulas. Time pressure sometimes pushes one toward better solutions!
Is the gap between 1 and 10 less than twice, equal to twice, or more than twice the gap between 1 and 3 ? Decide based on your understanding of ratios; then check your reasoning by measuring both gaps.
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