91Ó°ÊÓ

What kind of flux (flux of what?) is current density (current per area)?

In this problem, you use symmetry to maximize the gain of an \(L R C\) circuit or a spring-mass system with damping (using the analogy in Section 2.4.1). The gain \(G\), which is the amplitude ratio \(V_{\text {out }} / V_{\text {in }}\), depends on the signal's angular frequency \(\omega\) $$ G(\omega)=\frac{\frac{j \omega}{\omega_{0}}}{1+\frac{j}{Q} \frac{\omega}{\omega_{0}}-\frac{\omega^{2}}{\omega_{0}^{2}}} $$ where \(j=\sqrt{-1}, \omega_{0}\) is the natural frequency of the system, and \(Q\), the quality factor, is a dimensionless measure of the damping. Don't worry about where the gain formula comes from: You can derive it using the impedance method (Problem 2.22), but the purpose of this problem is to maximize its magnitude \(|G(\omega)| .\) Do so by finding a symmetry operation on \(\omega\) that leaves \(|G(\omega)|\) invariant.

On a logarithmic scale, how does the physical gap between 2 and 8 compare to the gap between 1 and 2 ? Decide based on your understanding of ratios; then check your reasoning by measuring both gaps.

Use Gauss's method to find the sum of the integers between 200 and 300 (inclusive).

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.

Use symmetry to find the missing coefficients in the expansion of \((a-b)^{3}\) : $$ (a-b)^{3}=a^{3}-3 a^{2} b+? a b^{2}+? b^{3} $$

On the logarithmic scale in the text, the gap between 2 and 3 is approximately \(1.85\) centimeters. Where do you land if you start at 6 and move \(1.85\) centimeters rightward? Decide based on your understanding of ratios; then check your reasoning by using a ruler to find the new location.

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) $$ \int_{0}^{\infty} \frac{\ln x}{1+x^{2}} d x . $$

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?

In an infinite grid of 1-ohm resistors, what is the resistance measured across one resistor? To measure resistance, an ohmmeter injects a current \(I\) at one terminal (for simplicity, imagine that \(I=1\) ampere). It removes the same current from the other terminal, and measures the resulting voltage difference \(V\) between the terminals. The resistance is \(R=V / I\). Hint: Use symmetry. But it's still a hard problem!