Prozorov_04

# Prozorov_04 - 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 4 Electric potential work along the path in a potential field o The work done raising a basketball against gravity depends only on the potential energy. It does not depend on other motions. A point charge moving in a field exhibits similar behavior. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 2 the work done moving a test charge As a test charge, q0, moves in the presence of another charge q. F d l Fdl cos Fdr b WA to B rb qq0 qq0 F d l k 2 dr k r r a ra rb ra qq0 qq0 k ke U b U a ra rb qq0 U ( r ) ke constant r PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 3 potential energy qq0 U (r ) k r o potential energy between like charges increases as the charges become close (repulsion). o Unlike charges have potential energy becoming negative as they become close (attractive). PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 4 The electrical potential (point charge) Let’s get rid of a test charge q0 and define: U r q V (r ) k q0 r Units: Volt 1 V = 1 J/C PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 5 general definition of electric potential B WA to B F d l U B U A , F q0 E A B WA to B q0 E d l q0 VB VA A B VB VA E d l A PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 6 differential definition electric potential is only defined up to some constant. It means that you could add (or subtract) ANY constant and all physics will remain the same. Why? – Because only potential DIFFERENCE has physical meaning! Why?, because we actually only need an electric field: F U F qE in electrostatics: E V U qV remember from 221: V V V ˆ ˆ E gradV V x y y z x PHYS222 - Prof. Ruslan Prozorov - Iowa State University ˆ z 29 August 2011 7 why? because . . . for ANY constant C: d f x C dx df x dC df x 0 dx dx dx so, we could add ANY constant to V to obtain the same electric field E V x, y, z V x, y, z C therefore we can write: V r E d l PHYS222 - Prof. Ruslan Prozorov - Iowa State University - indefinite integral defined up to a constant 29 August 2011 8 a charged ring let’s calculate V(r) along the xaxis. First we need electric field. 2 E x ka Q 2 a V x E d l kQ 0 x x2 a x x 2 2 3/2 a 2 3/2 dx d kQ kQ a2 x2 x x 2 a 2 3/2 C we can choose convenient constant C. For example, we can require that V(x) become zero at infinity. Then C must be zero. kQ V x 2 2 a x PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 9 an infinite line of charge 2k we already know electric field: E x dx x V x E d l 2k 2k ln x C where we placed constant C under ln, so that ln argument is dimensionless. We can choose ANY constant C and the potential diverges for x=0 and x=infinity. (There is nothing wrong with this as we only need the potential difference. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 10 an infinite charged plate we already know E sign x electric field: 2 0 sign x V x E d l dx 2 0 sign x sign x xC x 2 0 2 0 we can choose any C, so we used C = 0 PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 11 two parallel plates y what is the potential difference between two plates? ++++++++++++ E ------------- d x E 0 V E d l dy y C 0 0 V y y 0 (E points down – negative y – direction) Let’s choose C, so that V(y=0)=0 potential difference is: PHYS222 - Prof. Ruslan Prozorov - Iowa State University V d 0 29 August 2011 12 an electron in two parallel plates why is it useful? Because we control V by external source (a battery) of known voltage, so we do not need to know surface charge density, ++++++++++++ electron F E ------------- d dV F eE eV e dy V V y y 0 0 d V Fe e 0 d these are all known parameters PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 13 (old style) CRT computer monitor or TV Anodes (positive, accelerate electrons) Fluorescent screen Hot Electron cathode beams (emits electrons) PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 14 superposition principle we know that for an electric field, if we have several sources, the resulting field is a vector sum. Therefore, V E d l E Ei E d l E d l V i i i V Vi PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 15 Equipotential surfaces and field lines o Surfaces of equal potential are perpendicular to electric field. Their density is proportional to electric field strength PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 16 field lines and a conducting surface charge induced by polarization PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 17 potential in a conductor here, it is useful to use differential form. Inside a conductor: E V 0 this means that V CONSTANT usually, the constant is chosen by the boundary conditions, so that V is continuous across the boundaries. PHYS222 - Prof. Ruslan Prozorov - Iowa State University 29 August 2011 18 a conducting sphere inside: E 0, V C1 outside: q Ek 2 r V r E d l k q q dr k C2 r2 r Choose C1 and C2 to have V match at r = R. Obviously, C2=0 and C1 k inside: outside: PHYS222 - Prof. Ruslan Prozorov - Iowa State University q R q V k const R q V k r 29 August 2011 19 ...
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