Electricity and Magnetism Implications of Gauss s Law Electric Potential Energy

Electricity and Magnetism Implications of Gauss s Law Electric Potential Energy Lana Sheridan De Anza College Jan 22, 2018

Last time using Gauss s law

Overview implications of Gauss law electric potential energy

nducting slab placed he conductor must be Figure 24.16 A conducting Some Implications of Gauss s Law re not zero, free elec- slab in an external electric field S S S F 5 q Excess E) and Charge would on E. The a Conductor charges induced on the two surfaces of the slab produce ver, would We can mean argue that that an if electric an excess field that charge opposes is placed the on an isolated the existence of elecconductor. field of zero inside the slab. external field, giving a resultant conductor, that amount of charge will move entirely to the surface e the external of the conductor. field is None of the excess charge will be found within the conductor. the bodywhen of the conductor. to the left in Figure Gaussian the left surface. The surface e charge on the right field inside the cone, the surface charge agnitude of the intert field of zero inside equilibrium is on the instantaneous. nductor is also zero, within the conductor. rgue with the concept Figure 24.17 A conductor of arbitrary shape. The broken line.6. E = 0 inside the conductor, so the Gaussian surface shown cannot represents a gaussian surface conductor enclose in electro- a net charge. that can All be just excess inside charge the conduc- is on the surface.

Charges and Conductors Excess charge sits on the outside surface of a conductor. Close to the surface, the electric field lines are perpendicular to the surface. 1 Figure from OpenStax College Physics.

Some Implications of Gauss s Law23-7 APPLYING GAUSS L Faraday Ice Pail R R/2 (a) Gaussian surface + + + _ + _ + + _ + + _ + _ + + _ Fig. 23-11 (a) A negative point charge is located within a spherical the conductor. metal shell that is electrically neutral. (b) As a result, positive charge is nonuniformly distributed on the inner wall of the (E shell, = 0and foran the equal Gaussian amount surface of negative shown.) charge is uni- (b) _ + + + _ charge of 5.0 outer wall. The gested by Fig. 2 uniform becau skewed distribu not produce an ution of charge tive charges rep The field li approximately the shell and t shell the patte A charge placed inside a conducting shell appears on the outside skew of ofthe po the pattern is th and the shell w matter where in

Questions about applying Gauss s law What shape would you pick for the Gaussian surface if you wanted to find the electric field at a perpendicular distance r from a line charge of uniform charge density λ? (A) cube (B) rectangle (C) sphere (D) cylinder

Questions about applying Gauss s law What shape would you pick for the Gaussian surface if you wanted to find the electric field at a distance r from the center of a spherical conductor carrying a net charge? (A) cube (B) rectangle (C) sphere (D) cylinder

Some Implications of Gauss s Law Uniform Shell of Charge A shell of uniform charge attracts or repels a charged particle that is outside the shell as if all the shell s charge were concentrated at the center of the shell. If a charged particle is located inside a shell of uniform charge, there is no electrostatic force on the particle from the shell.

ositive Uniformsurface Sphere is drawn of concentric Charge ld at a a large, spherical gaussian with the sphere. For points inside the sphere, a spherical gaussian surface smaller than the sphere is drawn. s from lectric ection ld due eld for y inteemonhapter using a r a Q Gaussian sphere b r a Gaussian sphere Consider a uniform insulating sphere of charge, radius a, charge Figure 24.10 (Example 24.3) A uniformly charged insulating density ρ, total charge Q. sphere of radius a and total charge Q. In diagrams such as this one, the dotted line represents the intersection of the gaussian surface with the plane of the page. d uniibution center? How does the electric field strength change with distance from the s s law to find the electric field.

with the sphere. Uniform Sphere of Charge e electric field at a surface is drawn concentric For points inside the sphere, a spherical gaussian surface smaller than the sphere is drawn. oblem differs from s law. The electric scussed in Section electric field due ound the field for Chapter 23 by inteis example demonussions in Chapter electric field using a r a Q Gaussian sphere b E da = q enc Gaussian sphere Figure 24.10 (Example 24.3) A uniformly charged insulating ɛ sphere of radius a and total charge Q. In diagrams 0 such as this one, the dotted line represents the intersection of the gaussian surface Outside sphere with (for the plane r > of a): the page. is distributed unicharge distribution apply Gauss s law to find the electric field. 4πr 2 E = 1 ɛ 0 Q ymmetry, let s choose a spherical gaussian surface of radius r, concentric with the Q or this choice, condition E (2) = is satisfied everywhere on the surface and S? d S A 5 E da. E = ρr continued 4πɛ 0 r 2 E = k eq r 2 r a Inside sphere (for r < a): 4πr 2 E = 1 ( ) 4 ɛ 0 3 πr 3 ρ 3ɛ 0 = k eqr a 3

(2) E 5 Q / 3 pa Uniform 311/4pk Sphere e 2 r 5 of k Q e acharge 3 r 1for r, a2 rt (A). It shows that that would exist at e sphere. That is, if hysically impossible. ed from inside the e electric field from value from the outa E k E e Q a 3 r k E e Q r 2 a r Outside the sphere, Figure the electric 24.11 (Example field is the 24.3) same as for a point charge, strength Q, A located plot of E versus at ther for center a uniformly of the sphere. charged insulating sphere. The electric field inside the sphere (r, a) varies linearly with r. The field outside the sphere (r. a) is Inside the sphere, field varies linearly in the distance from the center and all charge outside the distance r cancels out!

Question +q according to their volume charge density, greatest first.the figure also shows a point P for each sphere, all at the same distance from the center of the sphere. (b) Rank the spheres according to the magnitude of the electric field they produce at point P,great- The figure shows four solid spheres, each with charge Q uniformly a c est first. distributed b through its volume. 3-22 Question 4. d P P P P σ (+) + + + + + + e σ ( ) 23-23 Question 5. Separation d 4d 9d iew All (a) (b) (c) (d) Fig. 23-25 Question 8. Rank the spheres according to their volume charge density, greatest first. The figure also shows a point P for each sphere, all at the same distance from the center of the sphere. (A) a, b, c, d (B) d, c, b, a (C) a and b, c, d (D) a, b, c and d 9 A small charged ball lies within the hollow of a metallic spherical shell of radius R.For three situations,the net charges on the ball and shell, respectively, are (1) 4q,0;(2) 6q, 10q;(3) 16q, 12q. Rank the situations according to the charge on (a) the inner surface of the shell and (b) the outer surface, most positive first. 10 Rank the situations of Question 9 according to the magnitude of the electric field (a) halfway through the shell and (b) at a point 2R from the center of the shell, greatest first. 1 Halliday, Resnik, Walker

Question +q the magnitude of the electric field they produce at point P,greatest first. The figure shows four solid spheres, each with charge Q uniformly a c distributed b through its volume. 3-22 Question 4. d Q uniformly distributed through its volume. (a) Rank the spheres according to their volume charge density, greatest first.the figure also shows a point P for each sphere, all at the same distance from the center of the sphere. (b) Rank the spheres according to P P P P σ (+) + + + + + + e σ ( ) 23-23 Question 5. Separation d 4d 9d (a) (b) (c) (d) Fig. 23-25 Question 8. Rank the spheres according to the magnitude of the electric field they produce at point P, greatest first. (A) a, b, c, d (B) d, c, b, a (C) a and b, c, d (D) a, b, c and d iew All 9 A small charged ball lies within the hollow of a metallic spherical shell of radius R.For three situations,the net charges on the ball and shell, respectively, are (1) 4q,0;(2) 6q, 10q;(3) 16q, 12q. Rank the situations according to the charge on (a) the inner surface of the shell and (b) the outer surface, most positive first. 10 Rank the situations of Question 9 according to the magnitude of the electric field (a) halfway through the shell and (b) at a point 2R from the center of the shell, greatest first. 1 Halliday, Resnik, Walker

Potential Energy What is the potential energy that a charge has due to the electric field of other charges in its vicinity?

Potential Energy Recall from 4A, there are many kinds of potential or stored energy: gravitational (U = mgh, or U = GMm r ) elastic (U = 1 2 kx 2 ) potential energy energy that a system has as a result of its configuration; stored energy; this always results from the action of a conservative force mechanical energy the sum of a system s kinetic and potential energies, E mech = K + U

Conservative Forces Conservative force acts on a part of the system such that following any closed path (one that ends back at the starting point) the work done on the system by the force is zero. Conservative forces are forces that do not dissipate energy. They conserve mechanical energy. For conservative forces, it is possible to define a potential energy function. W int = U where W int is the work done by the conservative force internal to the system and U is the change in the potential energy of the system.

from point to point, the from point to point, the electric potential energy of the gravitational potential energy of charge field Potential energy system change decreases. of a charge the q moving object field a distance system d? decreases. (Similar to lifting/lowering a mass.) Example: Potential Energy in a Uniform Field d d q m S E S g a U E = q Ed b U g = mgd

Summary implications of Gauss s law Faraday ice pail electric potential energy Homework Serway & Jewett: PREVIOUS: Ch 24, Section Qs: 25, 29, 31, 33, 39, 41, 43, 55, 61, 65 NEW: Ch 25, onward from page 767. Obj. Qs: 5, 9, 11; Concep. Qs: 3; Problems: 31, 33, 35, 55, 57