Electricity and Magnetism
# Circuit Behavior

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 \(\SI{420}{\ohm}\).

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

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 \(h= \SI{3}{\centi \meter}\).

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

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 \(100R\).

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.

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