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PHY2054 – Gauss’s Law page - 1 - Section 4.1 - GAUSS’ LAW Objectives to understand that the area of a flat surface can be represented by a vector to understand the concept of electric flux qualitatively and quantitatively to understand Gauss’ Law qualitatively HOMEWORK ASSIGNMENT: COMPLETE READING CHAPTER 18. 1.1 a. Obtain a sheet of graph paper and an index card. Determine the area of the graph paper and the index card. Is there a unique way to represent the area of a flat surface by a vector? Which direction would it point? How long would it be? Explain. b. Think about a sealed box. Is it possible to represent each surface of the box unambiguously by a vector? Explain your reasoning. Is it possible to represent a flat surface unambiguously by a vector? Explain. 1 PHY2054 – Gauss’s Law page - 2 - c. Write a rule for representing the area of a surface by a vector. Discuss your rule with the class (instructor will guide this). d. Roll the graph paper into a cylinder. Is it now possible to represent the area of the surface by a vector? Explain your reasoning. e. Could you represent the area of a small box on the graph paper by a vector when it is rolled into a cylinder? Explain. (Please do NOT proceed to or read the next pages until this activity is completed.) 2 PHY2054 – Gauss’s Law page - 3 - 1.2 The area of a flat surface can be represented by a single vector perpendicular to the plane of the surface. The length of the vector is proportional to the area. It is useful to define a quantity called electric flux. Electric flux is defined as a measure of the number of electric field lines passing through a surface. Consider a positive charge. The electric field lines all come out of the charge and then spread out. But how many lines should we use? Whatever we choose, the number should be proportional to the charge because we want all of this to make sense. So how many? Consider the following drawing. Let’s assume that there are N lines coming out from the charge q. We want N to be proportional to q. Think about this and provide a brief answer: must N be an integer? We know that if we consider a sphere of radius R, the Electric Field on the surface is the same everywhere and is given by Coulomb’s Law E 1 q2 . Consider a 4 0 R sphere drawn around the charge with a radius R. The number of lines that we count 2 should increase with both the area and the Electric Field. For the sphere A 4R . Consider a closed surface and define the flux as being positive if the lines are leaving the closed surface. So N~AE which for our sphere is 4R 2 1 2 q q . So the number of lines 4 0 R o leaving a closed surface is proportional to q as we wanted! What can you say if q is negative? Can we have a negative number of lines? Are they entering or leaving the surface (closed) if they exist at all? Prepare to discuss this. Let’s get back to our diagram: 3 PHY2054 – Gauss’s Law page - 4 - A B C There are three closed surfaces shown surrounding the same charge. How many lines are leaving each? (Count them.) A______________________ B _______________________ C ___________________ Going back to our applet: A 4 PHY2054 – Gauss’s Law page - 5 - Red represents the positive charges and blue the negative. The diameter of the charge is proportional to the charge and an advanced calculation is done to present these results. Count the number of lines leaving (positive) or entering (negative) each charge and verify that the number entering or leaving is proportional to the charge. Notice that one line may start on one charge and end on the other so it is entering or leaving depending on the surface that you construct. Consider the “closed” surface labeled “A”. How many lines are leaving the surface? Remember that a line entering is a “negative” line leaving. (Does this blow your mind??) How many lines are leaving the volume? How much charge is in the volume? ______ ______ Here is a simple two charge example where the charges are known: Draw a surface around the negative charge and around the positive charge and count the number of lines leaving these surfaces. Draw a surface that encloses BOTH charges and do the same thing. Fill in the following table: Surface Surrounds - chg Surrounds + chg Surrounds BOTH Lines Leaving Charge inside What do you conclude: 5 Lines/Charge PHY2054 – Gauss’s Law page - 6 - You have hopefully been able to state something close to Gauss’s Law: The number of lines leaving a closed volume is proportional to the charge enclosed. Instead of dealing with abstract “lines” it is more convenient to deal with the electric field itself and state Gauss’s law in a cleaner way. But you should have the idea via the previous exercises. The electric flux through a surface is defined as the magnitude of the electric field times the area of the surface times the cosine of the angle between the direction of the electric field and the area vector of surface EAcos The units of flux are Nm2/C. Qualitatively, flux is the number of field lines passing through a surface. When the angle between the area vector of the surface and the direction of the electric field is greater than 90, the flux is negative. 2.1 a. Consider the situations shown below. Each picture contains some charge and an imaginary box. In each case, determine if there is net flux through the box. The total electric flux through the imaginary box is the sum of the electric flux through each surface of the box. Remember that when the angle between the area vector of the surface and the direction of the electric field is greater than 90, the flux is negative. 6 PHY2054 – Gauss’s Law page - 7 - b. Qualitatively, what does the flux through the box depend on? When is there more flux and when is there less flux? When is there zero flux? Can you discuss your reasoning with the class? At this point your instructor will knit this together and show some applications. Gauss’ Law states that the net flux through an enclosed surface is proportional to the amount of charge enclosed by the surface. The constant of proportionality is 0. The value of 0 is 8.85 10-12 C2/ N m2. This can be written as net qenclosed 0 SUMMARY You should understand that the area of a flat surface can be represented by a vector. You should understand the concept of electric flux qualitatively and quantitatively and understand Gauss’ Law qualitatively. 7 PHY2054 – Gauss’s Law page - 8 - At this point you should be able to do the following problems: UNIT 4 EXERCISES 1) Consider the following boxes. The electric field vectors at the surface of the box are shown. Determine if there is a net flux through each box. Determine if there is charge inside each box. Explain how you know. (a) (b) (c) (pictures from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY, 1995) 8 PHY2054 – Gauss’s Law page - 9 - 2) (from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY, 1995) Here is a disk-shaped region of radius 2cm, on which there is a uniform electric field of magnitude 300 N/C at an angle of 30 degrees to the plane of the disk. Calculate the electric flux on the disk, and include the correct units. 3) (from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY, 1995) Here is a box on whose surfaces the electric field is measured to be horizontal and to the right. On the left face (3cm by 2cm) the magnitude of the electric field is 400 N/C, and on the right face the magnitude of the electric field is 1200 N/C. a) Calculate the electric flux on every face of the box. b) Calculate the total flux of the box. c) Calculate the total amount of charge that is inside the box. 9