Lecture 19

# Lecture 19 - Self Induction A solenoid(or indeed any...

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Self Induction •A solenoid (or indeed, any conductor) can generate its own Faraday effect EMF. •In a solenoid, B S =n µ 0 i=N µ 0 i/D, with N the number of turns and D the length. •Each turn sees a flux Φ B,1 =B S A, and there are N turns, so the total flux is Φ B =N Φ B,1 =( µ 0 N 2 A/D)i. •If I is not constant, then Φ B will change and a voltage difference will develop V S =-( µ 0 N 2 A/D)di/dt. •A similar effect occurs in any current-carrying conductor, with the generalization VL=-Ldi/dt. L is called the self- inductance, or often just the inductance. •For an air-core solenoid L S = µ 0 N 2 A/D.

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Units • Flux has units of Tm 2 , also called a Weber (Wb). • An old-fashioned term for B is “flux density” and you will see B still given in terms of Wb/m 2 sometime. • The unit of inductance is the Henry (H). The is our first “American” unit, named after Joseph Henry, who independently discovered magnetic induction.
Inductance of a toroid • An N turn toroid carrying current I with inner radius a, outer radius b, and height h has a field inside B T =N µ 0 i/2 π r. • One turn of the toroid will receive a flux contribution at a radius r of d Φ B =B T hdr=Nh µ 0 i/2 π ×dr/r. • Integrating gives, for one turn, Φ B,1 =Nh µ 0 i/2 π ×ln(b/a), and for N-turns, Φ B =N 2 hi µ 0 ln(b/a)/2 π . • We find the inductance by just dividing out the current, so L tor =N 2 h µ 0 ln(b/a)/2 π .

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Boosting Inductance • Let’s say this is a good thing. • For the solenoid and toroid, we can make the object bigger. This is often bad because of space constraints. • A sizable boost occurs by just increasing the number of windings, since L N 2 .
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Lecture 19 - Self Induction A solenoid(or indeed any...

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