Saturday, March 14, 2020
Proving the lens formula Essay Example
Proving the lens formula Essay Example Proving the lens formula Essay Proving the lens formula Essay When light passes from air to a denser material it slows down. In a concave lens the light has to travel further through the middle then through the sides. This has the affect of pushing the waves back in the middle and forward around the edge therefore effectively adding curvature to the wave. A similar thing happens when passing through a concave lens but obviously vice-versa, taking away curvature of the wave. The curvature that the lens adds or takes away is the Power of the lens, measured in dioptres. P=1/f, P is the power of the lens and f is the focal length. The focal length of a lens is the distance from a lens to its focal point, which is where the image of a distant object is formed. The shorter the focal length the more powerful the lens. The following formula is what I am going to attempt to prove that it is valid. It is used to give the focal length, and hence where the image is focused. 1/v+1/u=1/f Where v is the distance from the lens to its focal point, u is the distance from the object to the lens and 1/f is the power of the lens. This follows from the above, the power shows how much curvature is added to the wave. As a wave moves further away from an object the curvature of it decreases. This formula may also help me with my progress, as I can use it to calculate the magnification of the lens. m=v/u (when in focus) Where m is the magnification, v is the distance from the lens to the image and u is the distance from the lens to the object. The further away the image is from the lens the lower the magnification, and the closer the image is to the lens the higher the magnification. Hypothesis The lens formula for a convex lens valid. Apparatus I will use the following equipment during me experiment. A Convex lens and lens holder Small convex lens used to focus the image on the screen. Small plastic holder used to keep the lens in place. A Metre rule Will be used to measure both the object distance (U) and the image distance (V). It has millimetre units, although large and hard to take a precise measurement in mm. A 30 cm rule Will be used to increase the accuracy of the measurements, where the metre rule proves difficult to use to get an accurate result. A Small screen Flat white screen on a stand, used for the image to focus on. Image distance measured from the front of the screen. A Light source (Mains or Low Voltage) small Lamp, 40-watt bulb. Used to emit light and is part of the object. A wire mesh on a stand Used to create a clear, sharp focus-able image that I can take measurements from. Placed in front of the light source. Jack Webdale 02/05/2007 Page 2 Most of the apparatus I will use are self explanatory due to the background information. I will use the Wire mesh on a stand as part of the object, as I am satisfied that it will be easy to produce a clear, sharp image with it. If I was to use A light bulb with text printed on it, it may be harder to get a clear image as if I used ink, it may smudge or become blurred due to the heat of the lamp. I have also chosen to use a 30cm rule as well as a metre rule, as the metre rule may be inaccurate or cause problems when trying to measure to mm. A 30cm rule could be used to do this easily and to a better degree of accuracy. Prediction I would choose to predict that the formula is valid. Avoiding the fact that people have used the formula for years, especially opticians, and I gather it must work as they still use it! However, my task is to prove that it works. The diagram below shows what occurs when light travels through a converging lens. It shows where the focal points image and object distances are represented. Due to the proportions of the diagrams I have studied, it would seem logical that two reciprocals added together would produce a reciprocal which its decimal value would be less, which confirms the diagrams. Therefore, I believe that my experiment, if done accurately, should prove that the lens equation is true. Diagram of apparatus Consideration of the variables This Experiment relies heavily on accuracy of measurements and distances. Therefore, it is imperative that these are not affected while measurements are taken. I will take 2 measurements of Jack Webdale 02/05/2007 Page 3 Each required distance to ensure I should not have made an error. Ideally, if I would have time, I could do an average of results for one distance to ensure a reliable result each time. However, I feel checking twice, taking the distances where the image appears focused, and taking a middle value, will give reliable results. Another Variable that could affect the point where the image is focussed is exterior light not emitted from the object (lamp and mesh). Therefore, I will compose my experiment in a dark room, so that sunlight will not affect the image, and this should also help me to get a clearer focus of the image on the sheet. I will stick all apparatus to the bench when they are not being used, so that the non-variables are not affected or altered. Method Before I begin the real experiment, I have chosen to perform a preliminary experiment to discover a suitable range of distances I will get results from. I will also have an idea of the power of the lens, so I can judge its minimum and maximum distances to get a clear image on the sheet. To do this I will set-up the apparatus as shown in the diagram of apparatus, And I have chosen to make the Object and the lamp a constant position, due to the wires etc and difficulty of shifting it about all the time. Therefore, the Lens and the image sheet are the elements that I will move to focus the image. Preliminary research I began with a crude test, to get an approximate result for the focal length of the lens. I simply got a piece of paper, put it against a ruler, and with the lens; I placed it in front of a window, and focussed the image on the paper. I then had a measurement of approximately 15 cm. This would help me greatly in my experiment, as it would indicate immediately any results way off the mark, considering the variables and errors. I then also set up the apparatus as shown in the diagram, and used them to determine what distances I would use in my experiment. I placed a metre rule on the bench, and put on one end, the screen that the image would focus, and at the other, the object. I decided that I would not exceed this Object to Image (U V) distance, as measuring over a metre would cause problems, as I would need to have to metre rules, increasing the chance of making the results inaccurate. I then discovered how close I could put the lens to the object, until I could not get a focus on the screen (Where The focal length equals the object distance, and the image distance = ?). This was in between 13-18 cm for the object distance. Preliminary research Summery Part 1 * Collect all of the equipment displayed in the diagram of apparatus. * Check that the lamp works, and that the lens is not cracked or dented. * Place the lens on a flat surface, facing a window so that light from outside travels through the lens. Place the screen at the opposite end, so that it looks similar to the diagram above, but using the outside as the image. Move the screen until an image (Real, inverted) is focussed on the paper. * Measure the image distance, using a 30 cm ruler. (This is the approximate focal length). Part 2 * Collect all of the equipment displayed in the diagram of apparatus. * Check that the lamp works, and that the lens is not cracked or dented. * Set-up the equipment as shown in the diagram of apparatus, placing the object at one end of the metre rule, and the screen at the other (this is the maximum object to image distance). * Move the lens close to the object, and experiment with the lens and the screen until it is Jack Webdale 02/05/2007 Page 4 Impossible to get a focussed image on the screen. * Find the bounds which this occurs, and record the results (This is the area where the focal length is approximately the same as the object distance). There is no need for a table of results for this preliminary experiment, as not many results are taken, they are merely to give an idea of the expected results in the real experiment. Real Experiment * Collect all of the equipment displayed in the diagram of apparatus. * Check that the lamp works, and that the lens is not cracked or dented. * Set-up the equipment as shown in the diagram of apparatus, placing the object at one end of the metre rule, and the screen at the other (this is the maximum object to image distance). * Turn on the lamp; line it up with the mesh so that the lens, screen and the object line up against the metre rule. * Keeping the Screen and the Object stationary, move the lens up and down the metre rule until a clear, focused image of the mesh can be seen on the screen. * Measure the object distance to the lens, using the white mid-point line on the lens holder as a marker, then measure the Image distance, using the front of the screen as the marker. Use the 30cm rule so that the mm can be measured as accurately as possible. * The lens can move around 5 mm and still produce a clear and focussed image on the screen. This is merely due to the sensitivity of our eyes. For the following results, keep the object stationary, and move the screen 10 cm down the metre rule, decreasing the distance to the object each time. Measure the distances. For the Image distance, you will need to record two results, where the image beings to lose focus between the 5mm focus gap. These results can be used to obtain a midpoint, where the real focus is occurring. * Repeat this so 8 records have been taken. For each, be-aware of the results been recorded, and be aware that the focal length is approximately 15mm, and repeat any result that appears irregular. * As the experiment goes on, eventually a focused image will be impossible to obtain. This is where the image distance is equal to the focal length. You should not try and record results at this point and beyond. Table of results example My table will take this form: U (object distance) cm Min. V (Image distance) cm Max. V (image distance) cm Avg. V (image distance) cm 1/U + 1/V = 1/F F (Focal length) cm . . . . . . Risk Assessment All things considered, there are little risks presented with this experiment. I feel confident no special precautions need to be taken to ensure the safety of people partaking, or working near the experiment. The are few dangers which in extreme circumstances could cause a problem is the Light bulb. First because of the heat and the risk of burning a hand, which can be avoided by using a metal cover, not touching the bulb, and a cap over the cover to expose little of the lamp. The electricity supply could also be a danger, but I will ensure the wires are out of the way of tripping over, pulling the plug and causing any problems. Jack Webdale 02/05/2007 Page 5 The Second is the possibility of a dropped lens, leaving shards of glass on the floor, and in extreme circumstances these shards going into someones eye. To avoid this as much as possible, the lens will be placed in its holder, away from the edge of the bench. Also, if the lens is dropped, it is to be swept up immediately and a new lens to be used. The Results The experiment worked out sufficiently, although one result, where the object distance was 20.6cm, the Avg. V Distance was around 60.4 cm. I knew this must be an error due to the pattern of the decreasing V distance, and the Focal Length of this result would have worked out to be 15.361 cm, which is quite far out from the other results. Due to this I repeated the experiment for this result, using the same apparatus, which I had numbered in case this occurred. The Graphs of these results are on a separate sheet of graph paper. To be precise, I performed each measurement twice to ensure I hit the mark each time. The second measurement was basically a check for the first. I could not take results more than two decimal places for accuracy, as I merely used my own vision and judgement. If however, I had Specialised measuring equipment, this could have been more accurate. Evaluation and Conclusion For plotting the graph, I also needed the separate data for 1/V and 1/U. So I put them in the following table to allow me to successfully draw the graph. After analysing the graph, it can be seen that a strait line can be drawn through the points. This means that the Object distance (U) is Inversely proportional to the Image distance (V). Thus we can say that when any result is taken for say U, put in its reciprocal form, and then added to the reciprocal of V, the result is always the reciprocal of the focal length of the converging lens being used (discarding errors and inaccuracy in this statement). On my graph, it can be seen that the line doesnt travel exactly through every point, but in all cases travels through the error box. This shows that if the results were perfectly calculated without any chance of error, all the points would lie on a strait line. It also shows that although I conducted my experiment as accurately as possible, small errors did occur. With the graph, I have determined that the equation of the line is 1/v = (-1)1/u+1/f. From this statement the gradient of the line is always -1, and this is always the case wha tever the reciprocal of the focal length. Also, due to the -1 gradient, the X-axis intercept is also the reciprocal of the focal length. With the graph, I can determine the experiment was successful, as the straight line travels through both axis and at almost the same points. On the Y-Axis 1.167cm and on the X-axis 0.066cm. They both give a focal length of approximately 14.9cm Knowing that the focal length is approximately 15cm, I can conclude that my experiment was successful, and thus proves that the lens formula 1/U + 1/V = 1/F is valid. I decided not to put error bars on my graph, as I was not using the whole values of v and u, where I knew the errors spread over a 0.5cm distance for each measurement taken. However, even though I took middle values of v, it is still evident that errors took place. If I were to repeat the experiment, I would choose to take two values of u, the object distance as well as v then take the middle value. This may also reduce the chance of inaccuracy due to the human eye. There are little ways in which I could improve this experiment, except take many readings of a result, then take an average value. Doing this for every measurement taken, however, would be very time consuming, and if one reading happened to be far out, the average would not be that accurate.