Chapter 22: Gauss s Law

Transcription

1 Chapter 22: Gauss s Law How you can determine the amount of charge within a closed surface by examining the electric field on the surface. What is meant by electric flux, and how to calculate it. How Gauss s law relates the electric flux through a closed surface to the charge enclosed by the surface. How to use Gauss s law to calculate the electric field due to a symmetric charge distribution. Where the charge is located on a charged conductor.

2 Determination of Charge in a Closed Box In the previous chapter we have learned how we can determine the electric field due How can we determine the charge inside a closed box without opening it? Remember electrostatic forces are field forces doesn t need any We can examine the electric field around the box to have an idea about the charge/s inside the box. If the experienced electric field is outside Then the net charge enclosed should be (+)

3 Determination of Charge in a Closed Box Examples: How can we determine the charge inside a closed box without opening it? This is just an example to demonstrate that we have an idea about a charge distribution by choosing a box and studying the electric fields on the box. We will use this idea to determine the electric field of a symmetric charge distribution after defining the flux and the Gauss s Law.

4 Electric Flux and Enclosed Charge In the previous example, we have learned by looking at the electric field at the surface of a closed box, we qualitatively have an idea about the enclosed charge. Now we would like to look the problem from a flux point of view. Flux comes from Latin word flow. Electric flux is a measure of electric fields going through the surface. In terms of electric field lines, Electric flux related to number of field lines going through surface Now, let s study the flux through the enclosed surface (completely enclosing a volume) We need to look at 6 sides of the box to determine the net flux If the electric field is going outside means a + flux. If the electric field is going outside means a flux. Net Flux Total flux is the sum of fluxes from each surface. What is the electric flux through this box (enclosed surface); if there is no charge in it and it is located somewhere far in the space (no external field)?

5 Electric Flux and Enclosed Charge Electric Flux of a POSITIVE charge inside a box Because all the lines are outward = Positive Flux Electric Flux of a DOUBLED POSITIVE charge inside the same box Because the electric fields gets twice stronger = Flux doubles Conclusion: Flux is proportional with the enclosed net charge. Keep the charge same but DOUBLE the DIMENSIONS Electric field becomes smaller at the surface Because area now bigger still same amount of flux happens. *** SO FLUX REMAINS THE SAME *** Conclusion: Flux only depends on the enclosed net charge.

6 Electric Flux and Enclosed Charge No Charge No Fields No Flux Flux is zero Net Charge is Zero There are inward ( ) and outward fluxed (+) Net Flux will be zero No Charge There are inward ( ) and outward fluxed (+) Net Flux Flux is zero

7 Conclusion: Electric Flux and Enclosed Charge Whether there is a net outward or inward electric flux through a closed surface depends on the sign of the enclosed charge. If net charge is (+) (+) flux outward flux If net charge is ( ) ( ) flux inward flux Charges outside the surface do not give a net electric flux through the surface. whatever flux goes inward, the same amount goes outward. Net = 0 The net electric flux is directly proportional to the net amount of charge enclosed within the surface but is otherwise independent of the size of the closed surface or the shape of surface. What ever size or the shape flux is proportional with enclosed net charge

8 Page 728 In addition: What if we completely change the enclosed surface as like a closed sphere or cylinder instead of a box.

9 What is a Flux? Calculation What we actually mean by flux? The amount of something that goes through an area. = flow rate of Examples from everyday life: Sun light goes through the window == there is a light flux through windows Water flowing in a river == there is flux of water In electrostatics, we can also define the electric flux through a surface Measure of how much electric field lines are crossing the surface. But remember, field lines are not moving just they are in space so it is not like a flow but it can be defined as a flux.

10 Analogy: Fluid Flow Analogy How much water flux (volume / time) through a wire frame placed in a river? Assume that the water flows at constant (uniform flow) velocity everywhere FLUX dv va dt where, dv dt v A : Water volume per unit time (flux) : Speed of water : Area of the wire frame How much is the flux if frame is not perpendicular to velocity of the water? This time FLUX will be dv vacos dt Which can be written in the following vector form dv v A dt Mathematical definition of the electric flux is similar to equation of water flux (see next slide)

11 Flux of a Uniform Electric Field Perpendicular What is the flux through the surface A? A is lying in a perpendicular plane to E and E is uniform) Remember: Density of electric field lines tells us the magnitude of the electric field. Electric flux a measure of amount of lines passing through the surface is the product of the magnitude of the electric field and the surface area, A, perpendicular to the field Φ E = EA Scalar quantity Unit (N.m 2 )/C

12 Flux of a Uniform Electric Field not Perpendicular If the field is not perpendicular to the surface then we look at the amount of the vector those perpendicular to the surface nˆ Esin Consider an area A A constant electric field E passing through the area A E Ecos Area A Question: How much E. field is poking through (perpendicular to) the surface A Define a unit normal vector perpendicular to the surface Break the E into perpendicular and parallel components Only the normal component E n is passing through the surface Φ E = EA cos Can be also expressed as in scalar product of E and A=An vectors. E A E. Note: n unit vector is always in outside direction for a closed surface.

13 Examples: Flux of a Uniform Electric Field Determine the electric flux through surface A in terms? Remember the definition of flux: E A E.

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16 Flux of a Non uniform Electric Field Consider a small surface element where you may assume the electric field is constant Flux due this surface element d E d EdAcos E Total flux is obtained by considering the integration over all surface E surface E da

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19 Gauss s Law Gauss s law is a different form of Coulomb's Law. Both of them relates the electric field to the static charge. Either of them can be used to determine the electric field Gauss s law is very powerful in dealing with symmetric charge distributions. Coulomb s Law for charge: Gauss s Law states that Electric Flux through a closed surface is proportional to confined electric charge Electric Flux α Total Electric Charge E Qencl We will be using this idea to study the electric field due symmetric charge distributions Example: Electric Fields of a sphere, a line, a plate, or a cylinder

20 Gauss s Law : Point Charge Gauss s Law: Electric Flux α Total Electric Charge E Q encl Let s now apply Gauss s statement to simplest case, E field due to a point charge In order to keep it simple to calculate, consider an imaginary closed surface a sphere of radius, R and the point charge +q is located in the center. E = E da throughout Spherical Surface Because E is constant all through the spherical surface where E is the magnitude of E field and A is the surface area of the sphere Considering E q in = E da = E Q encl 0 General Form of Gauss s Law

21 Gauss s Law: Gauss s Law : General Form E = E da = The total electric flux through a closed surface is equal to the total (net) electric charge inside the surface, divided by ε 0. Examples: Q encl 0 Q encl = 0 no charge enclosed Q encl = +2q Q encl = +q q = 0 Φ E = 0 (zero flux no net flux) Φ E = 2q/ε 0 Φ E = 0

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23 Applications of Gauss s Law We will use how to use Gauss s law for the following purposes Inside a conductor E=0 then we will use this idea along with Gauss s law to determine charge distributions in conductors under various conditions with different geometries. 1. Charge destitution on a conduction objects (sphere, cylinder, etc.) 2. Electric field near the surface of a conductor. Gauss s law can be used to determine the electric field due to a given charge distributions with a nice symmetry to evaluate the integral easily. So it helps with calculating the electric field for a known charge distributions 1) Electric field of a charged sphere (inside or outside of the sphere) 2) Electric field of a charged infinitely long wire or a cylindrical cable/wire. 3) Electric field of a charged infinitely large plate/s

24 Excess Charges on a Conductor As we know, the are significant number of electrons can move freely on a conductor (~ %1) These charges will repel one another until the electrostatic balance is acquired inside the conductor electrons are not in motion anymore this is called static equilibrium. This happens once the electric field inside will be zero. In conclusion, electric field inside a conductor is always zero Now, let s use this idea to determine where the excess charges are located on/in a conductor

25 Page 737 * Electric field of conducting sphere as a function of radial distance.

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31 Charges on a Conductor Just as we have learned before, electric field inside a conductor is zero in static situation. We can use this knowledge along with the Gauss s law to determine the charge distribution on a conductor.

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33 Faraday s Cage

34 Electric Field at the Surface of a Conductor Right at or very near the surface of a conductor, electric field is perpendicular to the surface We can use the Gauss s law to determine the magnitude of the electric field

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49 RECOMMENDED END OF CHAPTER 22 QUESTIONS AND PROBLEMS 1,2, and 5 1,2,3,4,5,6,8,10,11,14,15,16,17,19,21,22,23,24,25, 26,28, and 34 34,38,40,42,43,45,46,47,51,52, and 53