Chapter 29 — Electromagnetic Induction

14 points · 3 extra credit

Chapter 29 — Electromagnetic Induction

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (1 pt •) A straight conducting rod of length $L=0.40\,\text{m}$ slides at constant speed $v=6.0\,\text{m/s}$ along two frictionless rails in a uniform magnetic field $B=0.25\,\text{T}$ perpendicular to the plane of the rails. Find the motional EMF induced across the ends of the rod.

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2. 2. (2 pts ••) A flat circular coil has $N=200$ turns and radius $r=3.0\,\text{cm}$. It sits with its plane perpendicular to a magnetic field that increases uniformly from $0$ to $0.50\,\text{T}$ in $0.020\,\text{s}$.

   (a) Find the magnetic flux through a single turn at the instant the field reaches $0.50\,\text{T}$.

   (b) Find the magnitude of the EMF induced in the coil while the field is changing.

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3. 3. (2 pts ••) For each situation, state the direction of the induced current in the loop and justify it with Lenz's law.

   (a) A circular loop lies flat on a table. A bar magnet is lowered toward it from above, north pole pointing down. Viewed from above, is the induced current clockwise or counterclockwise?

   (b) The same magnet is now pulled straight back up, away from the loop. Clockwise or counterclockwise (viewed from above)?

   (c) A rectangular loop moves at constant velocity entirely within a large uniform magnetic field (the loop never crosses a field boundary). Is there any induced current? Explain.

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4. 4. (2 pts ••) A square loop of side $a=0.15\,\text{m}$ sits in a uniform field $B=0.40\,\text{T}$. Initially the loop's normal is parallel to the field. The loop is then rotated so that its normal makes an angle of $60^\circ$ with the field, taking $0.10\,\text{s}$ to do so.

   (a) Find the magnetic flux through the loop before and after the rotation.

   (b) Find the average EMF induced during the rotation.

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5. 5. (3 pts •••) A conducting rod of length $L=0.50\,\text{m}$ and mass $m=0.020\,\text{kg}$ rides on horizontal frictionless rails of total circuit resistance $R=2.0\,\Omega$ in a vertical uniform field $B=0.60\,\text{T}$. A constant horizontal force $F=0.18\,\text{N}$ is applied along the rails, and the rod eventually reaches a terminal (constant) speed.

   (a) Explain physically why the rod approaches a terminal speed rather than accelerating forever.

   (b) Find the terminal speed $v_t$.

   (c) At terminal speed, find the induced current in the rod and verify that the electrical power dissipated equals the mechanical power delivered by the applied force.

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6. 6. (2 pts ••) A long solenoid of radius $R=6.0\,\text{cm}$ carries a current that is increasing so that the uniform field inside grows at $dB/dt=0.20\,\text{T/s}$. The changing field induces a circulating electric field.

   (a) Find the magnitude of the induced electric field at a radius $r=4.0\,\text{cm}$ from the axis (inside the solenoid).

   (b) Find the magnitude of the induced electric field at $r=10.0\,\text{cm}$ from the axis (outside the solenoid).

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7. 7. (2 pts ••) A parallel-plate capacitor with circular plates of radius $R=5.0\,\text{cm}$ is being charged. Between the plates the electric field is increasing at $dE/dt=2.0\times10^{12}\,\text{V/(m}\cdot\text{s)}$.

   (a) Find the displacement current between the plates.

   (b) Find the magnitude of the magnetic field between the plates at a radius equal to the plate radius, $r=R$.

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8. Extra Credit (3 pts •••)

   A flat rectangular loop of $N$ turns and area $A$ is placed in a uniform magnetic field of magnitude $B$. The loop is spun about an axis lying in its plane and perpendicular to $\vec{B}$, rotating at constant angular speed $\omega$. At $t=0$ the loop's normal is aligned with $\vec{B}$. This is the principle of the AC generator.

   (a) Starting from the definition of magnetic flux, write $\Phi_B(t)$ for one turn as a function of time.

   (b) Using Faraday's law, derive the induced EMF $\mathcal{E}(t)$ for the $N$-turn loop, and identify the peak EMF $\mathcal{E}_0$.

   (c) Explain, using your result, at what orientation of the loop the EMF is maximum and at what orientation it is zero — and why this makes physical sense in terms of the rate of change of flux (not the flux itself).

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