LECTURE 15 CONDUCTORS, ELECTRIC FLUX & GAUSS S LAW. Instructor: Kazumi Tolich

Transcription

1 LECTURE 15 CONDUCTORS, ELECTRIC FLUX & GAUSS S LAW Instructor: Kazumi Tolich

2 Lecture 15 2! Reading chapter 19-6 to 19-7.! Properties of conductors! Charge by Induction! Electric flux! Gauss's law! Calculating q using Gauss's law

3 Excess charge on a conductor 3! Since excess charge on a conductor is free to move, the charges will move so that they are as far apart as possible.! The excess charge on a conductor resides on its outer surface, as in diagram (a).! In (b), the charges are not as far apart as possible, so this cannot happen.

4 Demo: 1 4! Faraday ice pail! Demonstration of charge residing only on the outside of a conductor.

5 Electric field and electrons in conductors 5! If a conductor is placed in an electric field, electrons would be accelerated into a new location due to the electric force from that field.! Electrons would redistribute themselves until there is no electric field in the conductor.! A conductor shields its interior from external electric field.

6 Direction of E at the surface 6! The electric field is normal to the surface, otherwise the electrons would experience a force along the surface.! If the E field had a tangential component at the surface, the free charge would be accelerated until there is no electric force in the tangential directions.

7 Electric field and shape of a conductor 7! The electric field is stronger where the surface is more sharply curved.

8 Summary of properties of conductors 8! The E field is zero everywhere inside.! The excess charge resides on the surface.! The direction of E just outside is perpendicular to the surface.! The E field is larger near a sharp point.

9 Where should you hide during lightning? 9! During lightning, electrons travel in lightning bolts between clouds and Earth.! If a lightning bolt strikes a conductor, electrons would quickly (in the order of nano seconds) move to the surface, and E inside the conductor would be zero.! A car (unless it is a convertible) is almost an ideal conductive shell that you can hide inside!

10 Induction 10! If we bring a charged object next to a metal, the electrons in the metal will either! move towards a positively charged object.! move away from a negatively charged object.! Due to conservation of charge, this must leave a positive charge where the electrons have moved from.

11 11 Charging by induction

12 Demo: 2 12! Induction spheres! Demonstration of charging a conductor by induction

13 Electric flux 13! The amount of electric field penetrating a surface is called the electric flux.! If the direction of E that penetrates a flat surface with an area A, is constant, the flux is Φ = EAcosθ

14 Sign convention for electric flux 14! If the surface through with the flux is calculated is closed, the sign of flux is! positive for field lines that leave the closed volume of surface.! negative for field lines that enter the closed volume of surface.

15 Example: 1 15! A uniform electric field of magnitude E = N/C points upward. An empty closed shoe box has a dimension of d = 25.0 cm (depth) by w = 35.0 cm (width) by h = 20.0 cm (height). a) Which side has the greatest positive electric flux? Which side has the greatest negative electric flux? Which sides have zero electric flux? b) Calculate the electric flux through each of the six sides of the box.

16 Gauss's law 16! Gauss's law is given by Φ net = q enclosed ε 0 where ε 0 is the permittivity of free space, ε 0 = 1 4πk = C 2 N -1 m -2! The net flux through any closed surface is proportional to the net charge enclosed by that surface.! Gauss's law is one of the fundamental equations of electromagnetism (Maxwell s equations).

17 Gaussian surface 17! Gauss's law holds for any closed surface.! The closed surface used in Gauss's law is called the gaussian surface.! A gaussian surface is an imaginary surface that you construct.! If any two gaussian surfaces enclose the same charge, the number of field lines going out through each of them is the same, regardless of their shapes.

18 E in Gauss's law 18! The electric field E in Gauss's law is the total electric field due to all the charges both inside and outside the surface. Φ net = Φ 1 + Φ 2 + Φ 3! Since every electric field line due to q 3 enters and leaves the region bounded by the gaussian surface, Φ 3 is zero. q 2

19 Zero flux is not zero field! 19! There is zero flux through a closed surface even if an electric field exits on the surface if! there are charged particles outside, but no charged particles enclosed by the surface.! there are charged particles enclosed, but net charge inside the surface is zero.

20 20 Clicker question: 1

21 Example: 2 21! A single point charge is placed at the center of an imaginary cube that has 20-cm-long edges. The electric flux out of one of the cube s sides is Φ side = -1.5 kn"m 2 / C. How much charge is at the center?

22 Gauss s law and charges in conductors 22! Consider any arbitrary gaussian surface that resides completely inside a conductor.! E is zero everywhere on the gaussian surface.! The net flux through the gaussian surface is then also zero.! The charge enclosed by the gaussian surface is also zero according to Gauss s law.! Any net electric charge must resides entirely on the surface of the conductor.

23 Equivalence of Gauss s and Coulomb s laws 23! We can derive Coulomb s law for an electric field of a point charge using Gauss's law.! Consider a sphere of radius r around a point charge q. q! A point charge has a spherical symmetry; if the charge was rotated about the point where the charge is, you would not know that it was rotated.

24 Equivalence of Gauss s and Coulomb s laws 2 24! Based on the symmetry, we know that the electric field is also symmetric about that point.! The E field is therefore always normal to the red surface and has equal magnitude at all points on the surface. q Φ net = E4πr 2! From Gauss s law Φ net = q ε 0 E = 1 4πε 0 q r 2

25 Charge in a conducting shell 25 Charge at the center! Define a closed surface within the metal shell.! Because the electric field in the metal is zero, Gauss s law tells us that the total charge enclosed is zero.! There must be a negative charge with magnitude equal to the positive charge in the center induced on the inner surface of the metal.

26 Charge in a conducting shell 2 26! Define a closed surface outside the metal shell.! The total charge enclosed is equal to the positive charge in the center. Charge at the center

27 Charge in a conducting shell: 3 27 Charge at off-center! Now, the charge inside is moved away from the center.! The electric field inside the shell changes, and the charge distribution on the inner wall is not uniform.! The positive charge on the outer surface is uniformly distributed since the E field is 0 inside the shell, and perpendicular to the surface just outside.! But the electric field on the outside is equivalent to that from the charge being at the center.