PHY2049ch30D-31A%283-17-10%29

PHY2049ch30D-31A%283-17-10%29 - Induction & Inductance...

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Induction & Inductance (Chapt. 30 review & complete) Magnetic flux : B area BdA Φ= G G e.g. consider the magnetic field inside a solenoid where is uniform and a loop of area A. B G B BA B BAcos Φ B 0 Φ = x y z (units Tesla·m 2 = Weber ) θ B G dA G dA G dA G
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Faraday’s law of induction B d N dt Φ ξ=− Lenz’s law The EMF generated in a coil of N turns depends on the time rate of change of the flux. A changing magnetic flux through a coil of wire induces an EMF in the coil of wire. The direction of the EMF induced is such that the magnetic field produced by the current caused by the EMF acts to oppose the change in the flux (Lenz’s law).
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Faraday’s law (reformulated) B d Ed s dt Φ ⋅= G G v Even in the absence of a physical loop of wire a changing magnetic flux through a region of space generates an electric field that satisfies: The –s ign (Lenz’s law) sets the direction of relative to the direction of the integration and the changing flux. E G ds G The right side : , is the time rate of change of the enclosed magnetic flux. B d dt Φ The left side : , is a line integral over the resulting electric field for the closed path (indeed it is this field that causes the EMF & current when a physical wire is present). s G G v + E G B G (increasing inwards) E G E G Holds for any closed path.
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Example + B = 0 everywhere except for the circular region of radius r that contains a uniform magnetic field, into the page that is increasing as, o ˆ B(t) (B at)k =− + G ˆ k · For the closed path and direction shown determine . Ed s G G v B d s dt Φ ⋅= G G v Since, We need only evaluate the right side. Now, B area BdA Φ =⋅ G G The area in this integration is the area bounded by the closed path. This area is to the magnetic field so lies along . But there is a minor ambiguity as to whether they are parallel or anti–parallel. dA G B G
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We proceed by assuming them to be parallel. Then, + ˆ k · B area area BdA Φ= = ∫∫ G G B cirular area outside cirular of radius r area of radius r Bd A ( 0 ) d A + 2 B B( r ) π But so, o BB ( t )B a t = =+ 2 Bo (t) r (B at) π + Flux increases with time into the page. The time rate of change of this flux is then, 2 B d ra dt Φ Which gives the magnitude of the integral we seek. To find its sign we use the following considerations.
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Lenz’s law tells us that if there were a loop of physical wire, the current induced in the loop by the EMF would be in a direction to generate a B field that opposes this flux change. To create an outward pointing field the RHR requires a counter–clockwise current that would only be induced by a counter clockwise field. But the direction of integration taken in the problem is opposite this so for that direction we get, E G Since the flux is increasing inwards the induced field would point outwards .
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PHY2049ch30D-31A%283-17-10%29 - Induction & Inductance...

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