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Figure P15.49 shows a closed cylinder with a cross-sectional area . The constant electric field has a magnitude and is directed vertically upward, perpendicular to the cylinder's top and bottom surfaces so that no field lines paw through the curved surface. Calculate the electric flux through the cylinder's (a) top and (b) bottom surface, (c) Determine the amount of charge inside the cylinder.

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Kath
KathLv10
22 Nov 2020

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

A Gaussian surface in the form of a hemisphere of radius R = 5.40cm lies in a uniform electric field of magnitude E = 2.50 N/C. Thesurface encloses no net charge. At the (flat) base of the surface,the field is perpendicular to the surface and directed into thesurface.

(a) What is the flux through the base?
_____N·m2/C

(b) What is the flux through the curved portion of thesurface?
_____N·m2/C

Faraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. This problem is a prototypical example in which an increasing magnetic flux generates a finite line integral of the electric field around a closed loop that surrounds the changing magnetic flux through a surface bounded by that loop. A cylindrical iron rod with cross-sectional area is oriented with its symmetry axis coincident with the z axis of a cylindrical coordinate system as shown. It has a uniform magnetic field inside that varies according to  .

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1. Find  , the electromotive force (EMF) around a loop that is at distance R from the z axis, where R is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as positive

2. Due to the cylindrical symmetry of this problem, the induced electric field   can depend only on the distance R from the z axis, where R is restricted to the region outside the iron rod. Find this field

Learning Goal: To work through astraightforward application of Faraday's law to find the EMF andthe electric field surrounding a region of increasing flux

Faraday's law describes how electric fields and electromotiveforces are generated from changing magnetic fields. This problem isa prototypical example in which an increasing magnetic fluxgenerates a finite line integral of the electric field around aclosed loop that surrounds the changing magnetic flux through asurface bounded by that loop. A cylindrical iron rod withcross-sectional area is oriented with its symmetry axis coincidentwith the z axis of a cylindrical coordinate system as shown. It hasa uniform magnetic field inside that varies according to . In otherwords, the magentic field is always in the positive z direction,and it has no other components.

For your convenience, we restate Faraday's law here: , where is theline integral of the electric field, and the magnetic flux is givenby , where is the angle between the magnetic field and the localnormal to the surface bounded by the closed loop.
Direction: The line integral and surface integral reverse theirsigns if the reference direction of or is reversed. The right-handrule applies here: If the thumb of your right hand is taken along ,then the fingers point along . You are free to take the loopanywhere you choose, although usually it makes sense to choose itto lie along the path of the circuit you are considering.

Find , the electromotive force (EMF) around a loop that is atdistance from the z axis, where is restricted to the region outsidethe iron rod as shown. Take the direction shown in the figure aspositive.
Express in terms of , , , , and any needed constants such as , ,and .
Due to the cylindrical symmetry of this problem, the inducedelectric field can depend only on the distance from the z axis,where is restricted to the region outside the iron rod. Find thisfield.
Express in terms of quantities given in the introduction (andconstants), using the unit vectors in the cylindrical coordinatesystem,r^ ,theta^ , and z^ .

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