GAUSS LAW Ken Cheney ABSTRACT Gauss Law that "what goes in comes out" is checked for a number of sources of energy and flux including light, sound, nuclear radiation, and microwaves. The source geometries used are point sources, an "infinite" line, and an "infinite" plane. GENERAL The idea of conservation of energy, electric flux, magnetic flux, etc along with simple symmetry arguments makes it possible to analyze many situations that would otherwise lead to great computational problems. The first trick is to be convinced that indeed "what goes in comes out". This experiment compares the observed dependence of various kinds of flux on distance with the dependence predicted by Gauss s Law. EXPERIMENTS You should gather data on the dependence of intensity on distance for several sources. Be sure to include at least one point source, one "infinite" line source, and one "infinite" plane source. You should make a careful trade off between doing many sources quickly and doing a few sources very thoroughly. If you do many sources you will have more evidence of the generality of your results. If you do a few sources very well you will have more confidence in the results for the sources you did investigate. THEORY Gauss s Law says that the power or flux leaving a source is the same when measured through any closed surface enclosing the source. The surface can be simple and close to the source or complex and far away. In any case the same amount of power or flux must pass out of the surface as is produced by the source. We will use this principal to calculate the functional dependence of intensity on distance. INTENSITY We are usually interested in the intensity of the power or flux, say in watts per square meter. Intensity = Power Area The area of our detector will remain constant so the power received by the detector will be proportional to the intensity. FLUX Flux is a general term used in analogy to the rate of flow of a fluid through an area. Flux is often used for the integral of the electric field strength, E, through a surface: Flux = E ds LB1CGA.DOC 1.132, May 3, 2002 Ken Cheney 1
If you visualize the magnitude of E as being proportional to the density of electric field lines the analogy to a fluid flowing through a surface is quite close. By further analogy we can think of a flux of nuclear radiation, photons, magnetic field,... In many cases the flux is proportional to the power carried so we can just analyze the power flow. This is what we will do in the derivations here. Later you will do similar derivations with electric and magnetic flux. POINT SOURCE-SPHERICAL SYMMETRY Assume a point source is radiating a power P equally in all directions, spherical symmetry. By symmetry the power must be flowing perpendicular to any sphere we imagine centered on the source. The intensity I at any radius R then is given by the power divided by the area A of the sphere of radius R: Using A = 4πR 2 gives: This is the source of the familiar inverse square relationships. INFINITE LINE SOURCE I = P/A Point Source I = P 4πR 2 Equation 1 The power P is being radiated by an infinite line. Let: λ typically would have units of watts per meter. By symmetry the power is radiating radially outward, perpendicular to any cylinder that is aligned with the line source. No power will go through the ends of the cylinder. For a cylinder of height l and radius R the area A that the power goes through, the area of the sides of the cylinder, is given by: The power emitted by a segment of the line of length l is given by: λ= power unit length A = 2πRl P = lλ Intensity (I) equals Power over Area becomes: I = P A = lλ 2πRl Infinite Line I = λ 2πR Equation 2 2 GAUSS LAW Ken Cheney
Notice that this is not an inverse square relationship but is an inverse first power function. INFINITE PLANE SOURCE By symmetry the radiation from an infinite plane that is radiating uniformly is perpendicular to the plane. Let: σ= Power Area σ would typically be in units of watts per square meter. Since the radiation is all parallel the intensity does not decrease with distance. If we observe the radiation from a section of the plane of area A it will all flow through a similarly shaped surface of area A located any distance from the plane. Hence the intensity I is given by: Infinite plane source I =σ Equation 3 This is perhaps the most surprising result, the intensity is constant. This seems very strange and unbelievable until we remember that this is just the assumption we used for gravity when we wrote g = 9.8m/s. We know that gravity really is an inverse square force but if we are near the surface of the earth the earth appears to be an infinite flat plane and gravity appears to be constant. SOURCES - GENERAL CONSIDERATIONS POINT SOURCES The critical considerations for a point source are that it: 1. Be small relative to the distance to the detector. 2. Produce a reasonably uniform output in all directions. 3. Not have significant reflections from walls, tables, etc. Some sources that can satisfy these requirements are: 1. Small light, the intensity is measured with a photometer. 2. A source of nuclear radiation, the intensity is measured with a Geiger Counter. 3. A small speaker, the intensity is measured with a microphone. 4. The end of a long magnet. INFINITE LINE SOURCE The critical requirements here include the second and third above but the first requirement requires some more thought. The source will not really be infinitely long, or infinitely thin. If we measure too close to the source it may act like a plane, if we measure from too far it will tend toward a point. Some possible sources are: 1. An eight foot fluorescent bulb. 2. A long current carrying wire. GAUSS LAW Ken Cheney 3
INFINITE PLANE SOURCE We still must minimize reflections as in requirement 3. above but now both the first two requirements are difficult to meet. We don t have any infinite plane sources. The measurements must be made quite close to the plane sources we do have. As the distance becomes too big the source will begin to act like a point and the intensity will not be independent of distance. More difficult is the problem of uniform intensity. At best our light source is not very uniform. Possible sources are: 1. Frosted surfaces illuminated from the rear by lights. 2. The earth, gravity is the flux. Measure gravity at different heights. This is not entirely serious. MYSTERY SOURCE We have one mystery function, a microwave source with a small horn antenna. It will be upto youto analyze the behavior of intensity with distanceand decide on the most appropriate function to represent your results. CHARACTERISTICS OF INDIVIDUAL SOURCES AND DETECTORS LIGHT SMALL LIGHT SOURCE "INFINITELY LONG" LIGHT SOURCE "INFINITE" PLANE LIGHT SOURCE PHOTOMETER NUCLEAR RADIATION RADIOACTIVE SOURCE GEIGER COUNTER MAGNETISM - MAGNETIC POLE LONG MAGNET What we would like as a source is a North or a South magnetic pole. Unfortunately if such isolated poles exist they have not yet been found. We will have to make do with a North-South pair of poles that are separated by about one meter. This separation is produced by using a long thin magnet about a meter long. If we stay quite close to one of the poles its magnetic field will be much stronger than the field from the distant pole and we will have, in effect at least, an isolated pole. 4 GAUSS LAW Ken Cheney
If we get very far away from the pole we are working with however the other pole will begin to exert a significant effect and will confuse our results. COMPASS We will us a fairly ordinary compass as our magnetic field detector. A compass is a magnet mounted so it is free to rotate in the horizontal plane. The mounting should have as little friction as possible and may be rather fragile. Usually there is a way of raising the magnet off the delicate bearing while the compass is being moved. Be sure to unlock the clamp before using the compass and lock it again before returning the compass. Our compasses have a rather short magnet and a long pointer attached perpendicular to the magnet to increase the accuracy of angle readings. The compass should be level enough so the pointer doesn t scrape on the scale. THEORY OF MAGNETIC FIELD MEASUREMENT WITH LONG MAGNET We will find out how the strength of the magnetic field produced by our long magnet compares with the strength of the Earth s magnetic field. If you want the absolute value of the field due to the long magnet look up the strength of the horizontal component of the Earth s magnetic field at your location. The long magnetic is mounted vertically with the bottom end at the same height as the compass. Suppose we have our compass somewhere on a line directly east of one end of the long magnet. The magnetic field, B m, due to the long magnetic is in an East-West direction and the horizontal component of the Earth s field, B e is in a North-South direction. These two fields will add like vectors to produce the resultant field at the center of the compass. Suppose the compass pointer makes an angle θ to the East of North. The resultant magnetic field is in the direction θ measured from North. The resultant field is the hypotenuse of a triangle with the field from the long magnetic on the far side from the angle and the field of the earth on the near side. Hence: or tan(θ) = B m B e B m = B e tan(θ) B e Since is constant we can easily find how changes with distance. PROCEDURE FOR MAGNETIC FIELD MEASUREMENT WITH LONG MAGNET Mount the long magnet vertically with the bottom end at the same height as the compass. Use non magnetic rods and clamps to avoid producing extra magnetic fields. Of common materials only iron and steel are magnetic, aluminum, wood, and plastic are non magnetic. Use the compass to find the local magnetic North and mark out an accurate East-West line from the end of the long magnet. The compass will be moved along this line. B m GAUSS LAW Ken Cheney 5
Start with the compass as close to the end of the long magnet as possible and read off the direction of the resultant magnetic field at that location. Measure the angle from magnetic North, i.e. the direction the compass would point if the long magnet was removed. Record the distance from the end of the long magnet to the center of the compass. Repeat this measurement for ten or so location out to a distance of about a meter from the end of the long magnet. The last, outer, measurements will not be expected to be too good since the upper end of the magnet will be exerting a significant influence. ANALYSIS OF THE DATA FROM THE LONG MAGNET It is expected that the magnetic field strength from the end of the long magnetic will fall off inversely with the square of the distance from the magnet. If you don t have the facilities for doing a least squares fit you can plot the relative magnetic field strength versus 1/R 2 where R is the distance from the end of the long magnet to the center of the compass. Ideally this will give a straight line. Actually the close points should give a very good straight line while the outer points will begin to deviate due to the effect of the upper pole of the magnet. If you can do a least squares fit you could do a linear fit, y = A + B*x using y versus your calculated values for 1/R 2 for x. Or, if the program allows, use y = A + B/R 2 and give the actual R values for x. EXTRA POSSIBILITIES Dipol Field. Try putting the long magnet horizontally in an East-West direction and moving the compass along the bisector of the long magnet in a North-South direction. The magnetic field due to the long magnetic will be in the East-West direction and, at a great distance, should fall off inversely with the cube of the distance. Better perhaps, check out a small magnet so the field will not be too weak when you reach a distance large compared to the separation of the poles. MAGNETISM - LONG CURRENT CARRYING WIRE A straight current carrying wire produces a magnetic field that circles the wire. For an infinite wire the intensity of the field should fall off inversely with the distance. EQUIPMENT High current power supply (20a, Eico 1064S), 5-10m of heavy wire (zip chord or extension chord is fine), large compass THEORY OF MAGNETIC FIELD MEASUREMENTS WITH A LONG CURRENT CARRYING WIRE The principle is much the same as described above for a long magnet, the magnetic field strength from the long wire is compared to the magnetic field strength of the Earth. The geometry is slightly different. We want the field from the wire to be perpendicular to the Earth s field, so we place the magnet on a line going North-South through the current carrying wire. With this arrangement the discussion above "THEORY OF MAGNETIC FIELD MEASUREMENT WITH LONG MAGNET" applies very well. 6 GAUSS LAW Ken Cheney
POWER SUPPLY AND WIRE You want as large a current as possible to give a large, easily measurable magnetic field. The trick is to get this large current without melting the wire or power supply. It is fairly easy to check on the wire, just touch it occasionally to see it doesn t get too hot. You can increase the current in five amp increments while finding out how much current is safe for the wire. If you use too much current for the power supply it will probable blow its fuse. If the power supply can protect itself the teacher will probably tell you. Otherwise start with a small current and watch the ammeter on the power supply carefully as you increase the current. PROCEDURE FOR MAGNETIC FIELD MEASUREMENT WITH CURRENT CARRYING WIRE The wire should have as large a vertical section as possible, several meters if possible. The supports for the wire should be non magnetic: wood, aluminum, plastic... Be sure the wires from the power supply to the vertical wire are kept well away from the compass. All parts of the wire produce magnetic fields. Put the compass as near the center of the vertical wire as possible. Use the compass to find an accurate North-South line through the current carrying wire, with the current turned off. Move the compass along this line. Turn on the power supply very carefully. The wires have a low resistance and very little voltage is needed. Be sure to start with zero volts and go up slowly while you watch the ammeter. Start with the compass as close to the current carrying wire as possible. Read off the direction of the resultant magnetic field at that location. Measure the angle from magnetic North. Record the angle and the distance from the wire to the center of the compass. Repeat this measurement for an number of distances out to a maximum. The maximum is determined by either reaching a distance from the wire comparable to the length of the wire or finding that the magnetic field is too small to measure. ANALYSIS OF THE DATA FROM THE MAGNETIC FIELD OF A LONG WIRE It is expected that the field will fall off inversely with the distance from the wire to the compass. See the discussion above "ANALYSIS OF THE DATA FROM THE LONG MAGNET" and substitute 1/R for 1/R 2. SOUND SPEAKER MICROPHONE AND OSCILLOSCOPE GAUSS LAW Ken Cheney 7
MICROWAVES MICROWAVE GENERATOR MICROWAVE DETECTOR N.B. The intensity is proportional to the square of the voltage read on a voltmeter, not proportional to the voltage itself. ANALYSIS EXPECTED LEAST SQUARES CURVE FIT The experimental results of intensity vs. distance are to be compared with the theoretical relationship using a least squares curve fit program. The goodness of fit should be compared for several possible distance dependencies in addition to the expected one. A suggested program is FITS.PUB.PHYSCI on the HP 3000 mini computer at PCC. Using FITS it is very easy to try different functions. Suppose you have been using y = A + B/R 2 (obtained by answering "Function List:" by "1,1/X/X". To change to an inverse first power just do this, your typing is in bold face: X,Y--> NEW FUNCTION LIST: 1,1/X FITS will automatically do a Least Squares fit with the current data on the new function y = A + B/R. If the correlation coefficient and standard error look promising you can plot the new function, otherwise you can quickly go on to another function. Say y = A + Be R X,Y-->NEW FUNCTION LIST:1,EXP(-X) In seconds you will have a Least Squares fit to this new function. RANGE OF DISTANCE In the case of the "infinite" sources it is to be expected that the ideal behavior will only be observed within a limited range of distances. e.g. at sufficient distance the sources will begin to act as point sources. More surprising is that you can get too close to the "infinite" light source, it will begin to act like an "infinite" plane. The moral is that if your data looks strange at the extremes it may be trying to tell you something. Perhaps you should fit different parts of the data to different functions. WRITE UP EXPECTED A formal write up: Abstract, Introduction, Equipment Set Up, Plots and Least Squares Analysis, Analysis of data, Raw data, Conclusion. It is very important to have clear sketches showing the location and dimensions of the equipment and any surfaces around the equipment since reflections can easily invalidate the results. Be sure to discuss the physics of the results. i.e. do the experimental data match the expected functions of distance, if not can you guess why? Any evidence? 8 GAUSS LAW Ken Cheney