Coil 1 (base) NB Coil 2 (handle) NH a Figure 31.10 (Example 31.5) (a) This electric toothbrush uses the mutual induction of solenoids as part of its battery- charging system. (b) A coil of Ng turns wrapped around the center of a solenoid of N turns. O by Braun GmbH, Kronberg
An electric toothbrush has a base designed to hold the toothbrush handle when not in use. As shown, the handle has a cylindrical hole that fits loosely over a matching cylinder on the base. When the handle is placed on the base, a changing current in a solenoid inside the base cylinder induces a current in a coil inside the handle. This induced current charges the battery in the handle. We can model the base as a solenoid of length ℓ with NB turns (as shown), carrying a current i, and having a cross-sectional area A. The handle coil contains NH turns and completely surrounds the base coil. Find the mutual inductance of the system.
Given data,
Current =I
Number of turns in the handle = NH
Number of turns in the solenoid= NB
Length= L
Cross-sectional Area= A
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