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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

         

Figure 1 shows a cubic frame with side length \(s\). The frame is made of a material with magnetic permeability \(u\). Each arm of the frame has cross sectional area \(A\). Also, \(s^2 >> A\), and \(u >> u_\circ \), where \(u_\circ\) is the magnetic permeability of free space.

An \(N\) turn coil (\(N >> 1\)) is wound around each arm of the frame. Figure 2 shows, for all arms of the frame, how a coil is wound around an arm. Also in Figure 2, pay particular attention to the direction of the magnetic field produced by the current in the coil.

Figure 3 shows how the coils are electrically connected. The shaded dots indicate the terminals of the system.

The inductance of the system as seen across its terminals is \[ \dfrac XY u N^2 A/s , \]

where \(X\) and \(Y\) are coprime positive integers. Determine \(X+Y\).

Details and Assumptions:

  • In each arm, the magnetic field is uniform and also parallel to the arm.

  • The magnetic field does not leak to the outside of the material of the cubic frame.

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

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

The DC circuit above consists of two voltage sources \(V_1\) and \(V_2\) with internal resistances \(R_1 = 1 \Omega\) and \(R_2 = 2 \Omega\) respectively. There is a load \(R_L = 3 \Omega\) connected in parallel with the sources.

\(V_1\) and \(V_2\) are variable quantities.

Let \(P_1\), \(P_2\), and \(P_L\) be the amounts of power in watts dissipated by \(R_1\), \(R_2\), and \(R_L\) respectively.

Given that \(P_L = 10\) watts, determine the minimum possible value of \(P_1 + P_2 + P_L\) to 1 decimal place.

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/
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