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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 $$100R$$.

###### 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!

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