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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.

Level 4-5

         

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 \( \dfrac A{\omega- \phi} \).

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 \(B\).

Give your answer to 3 decimal places.

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/

A useless wire having a total resistance of \(48 \space \Omega\) is cut into 48 equal pieces. Then, a regular Deltoidal Icositetrahedron as shown below.

If the equivalent resistance between two opposite points, where four edges meet together is \(R \space \Omega\), then enter your answer as the value of \(100R\).


This question is part of the set Platonic Electricity.
Image credit: Wikipedia

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 \(a + b\).

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