Prozorov_03 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 3 Gauss’s law a concept of FLUX • here flux is the amount of water that flows through the rectangle per second. vt Vol Area vt , t 1 • If we considered flux through a rectangle, the flux will change as the rectangle changes orientation to the flow. Vol cos Area v PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 2 flux in a uniform electric field E E A EA cos PHYS222 - Prof. Ruslan Prozorov - Iowa State University N 2 m Vm C 26 August 2011 3 flux through the disk E 36.755 V/m E E A E r cos 36.755 3.14 0.1 cos 2 PHYS222 - Prof. Ruslan Prozorov - Iowa State University 2 6 1 Vm 26 August 2011 4 flux as the flow out of an imagined box • If we construct a boundary around a charge, we can think of the flow of electric field coming out from the charge like water through a screen surrounding a sprinkler. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 5 changing the conditions PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 6 no charges – no NET flux each line that enters an object must exit, so unless new lines are created inside, total (net) flux will be zero PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 7 effect of changing the sign of the charge PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 8 Gaussian surfaces PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 9 non – uniform field of point charge d E EdA cos E d E EdA cos we know that: therefore, q Er k 2 r q q q 2 E k 2 dA k 2 4 r r r 0 does not depend on the radius! PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 10 Flux through concentric spheres with different radii • Consider the flux as changing the radius of the sphere changes its volume. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 11 projecting flux through other shapes To calculate flux • only need area perpendicular to the field lines • we can then scale that to map on a sphere PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 12 Gauss’s Law E E dA surface Q 0 - integral form Gauss’s theorem: E dA surface Carl Friedrich Gauss (1777–1855) EdV volume C Q dV , - charge density 3 m volume E 0 PHYS222 - Prof. Ruslan Prozorov - Iowa State University - differential form 26 August 2011 13 practical applications: conductors In a conductor, charges can only be on the surface (they try to move as far from each other as possible). Gauss’s theorem tells us that in such case, electric field inside a conductor is always zero. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 14 charges on conductors PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 15 electric field of a line charge there is total charge q=l inside the Gaussian surface. l Total flux through the surface is: E 2 rlE 0 0 1 Therefore: E 2 0 r Q PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 16 electric field of a plane charge there is total charge q=A inside the Gaussian surface. A Total flux through the surface is: E 2 AE 0 0 Therefore: E 2 0 Q PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 17 electric field between parallel plates PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 18 insulating uniformly charged sphere inside: imagine a Gaussian sphere of radius r. It contains total charge q. E r E r Ar q r 0 4 3 r 3 q r r Q E r k 3r Ar 0 3 0 R A r 4 r 2 , q r outside: E r E r Ar Q 0 Q Q A r 4 r , E r k 2 Ar 0 r 2 PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 19 insulating uniformly charged cylinder inside: imagine a Gaussian sphere of radius r. It contains total charge q per length L of the cylinder. r R E r E r Ar q r 0 A r 2 rL, q r r 2 L q r r 2 E r k 2 r Ar 0 2 0 R E outside: E r E r Ar R r PHYS222 - Prof. Ruslan Prozorov - Iowa State University Q 0 Q 2 A r 2 rL, E r k Ar 0 r 26 August 2011 20 Experimental tests of Gauss’s Law PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 21 a Faraday cage blocks electric field PHYS222 - Prof. Ruslan Prozorov - Iowa State University 26 August 2011 22 ...
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