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# Circuit Behavior

Any circulating flow constitutes a circuit. Learn how to model the logic boards in your computer, the flow of nutrients in your blood, or the daily fluctuations in the temperature of your house.

# Circuit Behavior: Level 4-5 Challenges

You are given a disk of thickness $$h$$ with inner and outer radii $$r_1$$ and $$r_2$$, respectively. If the resistivity of the disk varies as $$\rho = \rho_0 \left|\sec \varphi\right|$$, where $$\varphi$$ is the polar angle, find the resistance between the points $$A$$ and $$B$$.

Details and Assumptions:

• The inner and outer rims are metal rings with zero resistance.
• Take $$\dfrac {r_2}{r_1} = e^2 \approx 7.389$$, $$\rho_0 = \SI{10}{\ohm \meter}$$, and $$h= \SI{3}{\centi \meter}$$.

You have each of the five Platonic solids, with edges made out of $$1\Omega$$ resistors. For which solid can you pick two vertices that have an effective resistance between them that is greater than $$1\Omega$$?

Definitions:

• Tetrahedron is a polyhedron with 4 sides.

• Octahedron is a polyhedron with 8 flat faces.

• Dodecahedron is a polyhedron with 12 flat faces.

• Icosahedron is a polyhedron with 20 flat faces.

• Oblate spheroid is a quadric surface obtained by rotating an ellipse about its minor axis.

Clarification: "All of these" and "None of these" refer only to the geometrical answers not to each other!

###### Image credit: http://pixelstains.net/

There are $$n + 1$$ resistors in the upper and lower rows, respectively, and $$n$$ resistors between the rows.

The value of the total resistance between A and B can be represented as $$(a \times 2^n - b)R$$, where $$a$$ and $$b$$ are positive constants.

Find $$a + b$$.

In the circuit above, wire $$AB$$ has length $$40\text{ cm}$$ and resistance per unit length $$\SI[per-mode=symbol]{0.5}{\ohm\per\centi\meter}$$. The voltmeter is ideal.

If we want to make the reading in the voltmeter vary with time as $$V(t) = 2 \sin(\pi t) \ \si{\volt},$$ then what should be the velocity of the contact (the arrow-tipped end of the wire above) as a function of time?

If the velocity can be expressed as $$A\sin(\omega t+\phi) \text{ cm}\,\text{s}^{-1},$$ where $$0<\phi<\pi$$, then enter the value of $$\dfrac A{\omega- \phi}$$.

Suppose there is a box containing an ideal sinusoidal AC voltage source (RMS magnitude of $$V$$) in series with a resistance $$R$$ and an inductive reactance $$X$$ (for a total complex impedance $$Z = R + jX)$$. The circuit is incomplete, and a pair of terminals are brought outside the box. Aside from the availability of the terminals, the box is closed and its contents are inaccessible.

You want to find out the values of the resistance and inductive reactance, so you devise a clever strategy to determine them. You take three measurements, each time connecting an ideal AC ammeter in series with a test impedance $$Z_T$$ and connecting the series combination across the box terminals.

For the first measurement, $$Z_T$$ is a perfect short-circuit with zero impedance, and the measured RMS current magnitude is $$\Large{\frac{V}{\sqrt{\alpha}}}$$.

For the second measurement, $$Z_T$$ is a $$1 \Omega$$ resistance, and the measured RMS current magnitude is $$\Large{\frac{V}{\sqrt{\alpha+5}}}$$.

For the third measurement, $$Z_T$$ is a $$1 \Omega$$ inductive reactance, and the measured RMS current magnitude is $$\Large{\frac{V}{\sqrt{\alpha+9}}}$$.

What is the value of $$R + X$$?

Note: In electrical engineering, the letter $$''j''$$ is commonly used to denote $$\sqrt{-1}$$, since the letter $$''i''$$ is commonly used to represent current.

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