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# assignment1 - ωt α)ˆ z Find the induced current density...

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Classical Electrodynamics II Prof. Hyun-Chul Kim Assignment 1: Faraday’s Law Due 12:00, Thursday March 17, 2010 Problem 1. A long solenoid of radius a , carrying n turns per unit length, is looped by a wire with resistance R , as shown in Fig. 1. 1: Problem 1 1. If the current in the solenoid is increasing at a constant raite ( DI/dt = k ), what current ﬂows in the loop, and which way (left or right) does it pass through the resistor? 2. If the current I in the solenoid is constant but the solenoid is pulled out of the loop, turned around, and reinserted, what total charge passes through the resistor? Problem 2. A spherical shell of radius a rotates about the z axis with constant angular velocity ω . It is in a uniform magnetic ﬁeld which is in the xz plane and at an angle α with the axis of rotation. Find the induced electric ﬁeld at each point on the sphere. Problem 3. A very thin conducting disc of radius a and conductivity σ lies in the xy plane with the origin at its center. A spatially uniform magnetic ﬁeld is also present and given by B = B 0 cos(

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Unformatted text preview: ωt + α )ˆ z . Find the induced current density J f produced in the disc. Problem 4. 1. Use the analogy between Faraday’s law and Amp` ere’s law, together with the Biot-Savart law, to show that E ( r , t ) =-1 4 π ∂ ∂t ∫ B ( r ′ , t ) × ( r-r ′ ) | r-r ′ | 3 dτ ′ , (1) for induced electric ﬁelds. 1 2. Show that E =-∂ A ∂t , (2) where A is the vector potential. Check this result by taking the curl of both sides. Problem 5. An electromagnetic “eddy current” brake consists of a disc of conductivity σ and thickness d rotating about an axis passing through its center and normal to the surface of the disc. A uniform B is applied perpendicular to the plane of the disc over a small area a 2 located a distance ρ from the axis. Show that the torque tending to slow down the disc at the instant its angular speed is ω is given approximately by σωB 2 ρ 2 a 2 d . Problem 6. Griﬃths, Problem 7.7. Problem 7. Griﬃths, Problem 7.23. 2...
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assignment1 - ωt α)ˆ z Find the induced current density...

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