Consider two cylindrical coils placed within one another. The outer coil has a radius of r₁, and n₁ turns/unit length. The inner coil has a radius of r₂ with n₂ turns/unit length. Find the mutual inductance per unit length between the two cylindrical coils.

Tests on an inductor show that a self-induced emf of magnitude 0.185 V is generated when the current is increased at a rate of 0.580 A/s. The manufacturer's label shows that the inductor has 550 turns. Determine the mean magnetic flux in each turn when the inductor draws a current of 0.770 A.

Analysis shows that the current across a 15 mH induction coil changes from 180 mA to 80 mA in a duration of 20 μs. Calculate the potential difference across the inductor, and specify if the potential difference increases or decreases in the direction of the current.

A circuit is constructed from resistors and an inductor as follows. The current at t = 0 is I₀. Calculate the time after t = 0 s when the current is (1/4)I₀.

A physics student sets up an arrangement with 2 resistors and an inductor connected to an ideal battery, as depicted in the diagram. The student kept the switch open for a prolonged duration and subsequently closed it at t=0s. After the switch has been closed for a prolonged period of time, determine the current flowing through the 35 Ω resistor.

An advanced sensor detects nanoscale gaps using inductance. Consider an LC circuit with a parallel-plate capacitor having plate area A and known inductance L. The charge oscillates at a frequency f when the capacitor plates are separated by a distance x. The relationship between 𝑥, 𝜖₀, 𝐴, 𝑓, and 𝐿 is given by: x=4π²Aϵ₀f²L.

When the plate separation changes by a small amount ∆x, the oscillation frequency changes by ∆f. Using the given relationship, derive an expression approximating ∆x/x.

An advanced sensor is used to detect gaps in the nanoscale using inductance. Consider forming an LC circuit using a parallel-plate capacitor with plate area 𝐴 and a known inductance 𝐿. If the charge is found to oscillate in this circuit at a frequency 𝑓=𝜔/2𝜋 when the capacitor plates are separated by distance 𝑥, derive an expression for 𝑥 in terms of 𝜖₀, 𝐴, 𝑓, and 𝐿.

A series LRC circuit is built from an inductor, L = 0.800 H, and a capacitor, C = 4.00 μF. Determine the resistance of a resistor that will cause critical damping.