Circuit Behavior: Level 4-5 Challenges

         

8 resistors (orange color) are connected to form a regular octagon. 8 more resistors (blue color) connect the vertices of the octagon to its center. All the 16 resistors are of resistance 420 Ω\SI{420}{\ohm}.

If the connecting wires have negligible resistance, calculate the equivalent resistance (in ohms, rounded to the nearest integer) between the terminals AA and BB.

You are given a disk of thickness hh with inner and outer radii r1r_1 and r2r_2, respectively. If the resistivity of the disk varies as ρ=ρ0secφ\rho = \rho_0 \left|\sec \varphi\right|, where φ\varphi is the polar angle, find the resistance between the points AA and BB.

Give your answer to 3 decimal places.

Details and Assumptions:

  • The inner and outer rims are metal rings with zero resistance.
  • Take r2r1=e27.389\dfrac {r_2}{r_1} = e^2 \approx 7.389, ρ0=10 Ωm\rho_0 = \SI{10}{\ohm \meter}, and h=3 cmh= \SI{3}{\centi \meter}.

In the circuit above, wire ABAB has length 40 cm40\text{ cm} and resistance per unit length 0.5 Ω/cm\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)=2sin(πt) V, 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 Asin(ωt+ϕ) cms1,A\sin(\omega t+\phi) \text{ cm}\,\text{s}^{-1}, where 0<ϕ<π0<\phi<\pi , then enter the value of Aωϕ \dfrac A{\omega- \phi} .

A useless wire having a total resistance of 48 Ω48 \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 ΩR \space \Omega, then enter your answer as the value of 100R100R.


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

The DC circuit above consists of two voltage sources V1V_1 and V2V_2 with internal resistances R1=1ΩR_1 = 1 \Omega and R2=2ΩR_2 = 2 \Omega respectively. There is a load RL=3ΩR_L = 3 \Omega connected in parallel with the sources.

V1V_1 and V2V_2 are variable quantities.

Let P1P_1, P2P_2, and PLP_L be the amounts of power in watts dissipated by R1R_1, R2R_2, and RLR_L respectively.

Given that PL=10P_L = 10 watts, determine the minimum possible value of P1+P2+PLP_1 + P_2 + P_L to 1 decimal place.

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