30. Induction and Inductance

An outer solenoid with 30 turns is wound tightly around an inner coil 25cm long with a diameter of 4cm and 300 turns. The current in the inner solenoid is 0.12 A and is increasing at a rate of 1.75×10³A/s. a) What is the average magnetic flux through each turn of the outer coil? b) If the resistance of the outer coil is 20mΩ, what is the magnitude of the induced current through the outer coil?

(Note:for the multiple choice, select the correct answer for part (b) only.)

CALC High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE CP30.86. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius r₁ while the outer conductor of radius r₂ is grounded. A soft, flexible insulating material fills the space between them, and an insulating plastic coating goes around the outside. Evaluate the inductance per meter of a cable having r₁ = 0.50 mm and r₂ = 3.0 mm.

(II) Determine the mutual inductance per unit length between two long solenoids, one inside the other, whose radii are r₁ and r₂ (r₂ < r₁) and whose turns per unit length are n₁ and n₂.

Two coils have mutual inductance M = 3.25 × 10^-4 H. The current i₁ in the first coil increases at a uniform rate of 830 A/s. What is the magnitude of the induced emf in the second coil? Is it constant?

(II) A 2.44-m-long coil containing 265 loops is wound on an iron core (average μ = 1850μ₀) along with a second coil of 115 loops. The loops of each coil have a radius of 2.00 cm. (a) Determine the mutual inductance M. (b) If the current in the first coil drops uniformly from 12.0 A to zero in 98.0 ms, determine the emf induced in the second coil.

A small thin coil with N₂ loops, each of area A₂, is placed inside a long solenoid, near its center. The solenoid has N₁ loops in its length 𝓁 and has area A₁. Determine the mutual inductance as a function of θ, the angle between the plane of the small coil and the axis of the solenoid.

(III) A long straight wire and a small rectangular wire loop lie in the same plane, Fig. 30–25. Determine the mutual inductance in terms of 𝓁₁, 𝓁₂, and w. Assume the wire is very long compared to 𝓁₁, 𝓁₂, and w, and that the rest of its circuit is very far away compared to 𝓁₁, 𝓁₂, and w.