The Index of Refraction and the Speed of Light The Index of Refraction and the S The Index of Refraction and the Speed of Light Key Idea

The Index of Refraction and the Speed of Light

Key Idea

  • Index of Refraction (n): A distinct optical property of a material that measures how much light changes speed and bends when passing through.

Definition

  • The index of refraction for a material is defined as: n=cvn = \frac{c}{v} Where:

    • c = speed of light in a vacuum (always 3.00×1083.00 \times 10^8 m/s)

    • v = speed of light in the material

Speed of Light in a Vacuum

  • The speed of light in a vacuum is:

    • c=300,000,000c = 300,000,000 m/s (3.00 \times 10^8 \text{ m/s}</p></li></ul></li><li><p>Noknownobjecttravelsfasterthanlightinavacuum.</p></li><li><p>Thespeedoflightinanyothermaterialisalwayslessthan</p></li></ul></li><li><p>No known object travels faster than light in a vacuum.</p></li><li><p>The speed of light in any other material is always less than3.00 \times 10^8m/s.</p></li></ul><h4id="f5fbb612−1aee−47dc−94f8−4c2d744009c9"data−toc−id="f5fbb612−1aee−47dc−94f8−4c2d744009c9"collapsed="false"seolevelmigrated="true">IndexofRefractionofVariousMedia</h4><ul><li><p>Thefollowingtableshowstheindexofrefraction(n)forvariousmedia:</p><ul><li><p><strong>Air/Vacuum</strong>:m/s.</p></li></ul><h4 id="f5fbb612-1aee-47dc-94f8-4c2d744009c9" data-toc-id="f5fbb612-1aee-47dc-94f8-4c2d744009c9" collapsed="false" seolevelmigrated="true">Index of Refraction of Various Media</h4><ul><li><p>The following table shows the index of refraction (n) for various media:</p><ul><li><p><strong>Air/Vacuum</strong>:n = 1.00</p></li><li><p><strong>Water</strong>:</p></li><li><p><strong>Water</strong>:n = 1.33</p></li><li><p><strong>EthylAlcohol</strong>:</p></li><li><p><strong>Ethyl Alcohol</strong>:n = 1.36</p></li><li><p><strong>Quartz</strong>:</p></li><li><p><strong>Quartz</strong>:n = 1.46</p></li><li><p><strong>VegetableOil</strong>:</p></li><li><p><strong>Vegetable Oil</strong>:n = 1.47</p></li><li><p><strong>Acrylic</strong>:</p></li><li><p><strong>Acrylic</strong>:n = 1.49</p></li><li><p><strong>Glass</strong>:</p></li><li><p><strong>Glass</strong>:n = 1.52</p></li><li><p><strong>Zircon</strong>:</p></li><li><p><strong>Zircon</strong>:n = 1.92</p></li><li><p><strong>Diamond</strong>:</p></li><li><p><strong>Diamond</strong>:n = 2.42</p></li></ul></li></ul><h4id="29e1c6f3−bdb0−4893−ab04−590b6767ed16"data−toc−id="29e1c6f3−bdb0−4893−ab04−590b6767ed16"collapsed="false"seolevelmigrated="true">UnderstandingtheIndexofRefraction</h4><ol><li><p><strong>LightSpeedinGlassvs.Air</strong></p><ul><li><p>Lighttravelssloweringlassthaninair.</p></li><li><p><strong>ExpectedSpeed</strong>:Sloweringlass;</p></li><li><p><strong>Indices</strong>:</p></li></ul></li></ul><h4 id="29e1c6f3-bdb0-4893-ab04-590b6767ed16" data-toc-id="29e1c6f3-bdb0-4893-ab04-590b6767ed16" collapsed="false" seolevelmigrated="true">Understanding the Index of Refraction</h4><ol><li><p><strong>Light Speed in Glass vs. Air</strong></p><ul><li><p>Light travels slower in glass than in air.</p></li><li><p><strong>Expected Speed</strong>: Slower in glass;</p></li><li><p><strong>Indices</strong>:n{air} = 1.00,,n{glass} = 1.52</p></li></ul></li><li><p><strong>LightSpeedinZirconvs.Diamond</strong></p><ul><li><p>Lighttravelsfasterinzirconthanindiamond.</p></li><li><p><strong>Indices</strong>:Zircon’sindexislessthandiamond’s.</p></li></ul></li><li><p><strong>ObservationofIndexValues</strong></p><ul><li><p>Allvaluesofninthetablearegreaterthan1.</p><ul><li><p>Thisisbecausethespeedoflightinanymaterial(v)isalwayslessthanthespeedoflightinvacuum(c),thus:</p></li></ul></li><li><p><strong>Light Speed in Zircon vs. Diamond</strong></p><ul><li><p>Light travels faster in zircon than in diamond.</p></li><li><p><strong>Indices</strong>: Zircon’s index is less than diamond’s.</p></li></ul></li><li><p><strong>Observation of Index Values</strong></p><ul><li><p>All values of n in the table are greater than 1.</p><ul><li><p>This is because the speed of light in any material (v) is always less than the speed of light in vacuum (c), thus:c > vimpliesimpliesn > 1.</p></li></ul></li><li><p>Ifamaterialwasdiscoveredwherelighttraveledfasterthanthroughair/vacuum,theindexofrefractionwouldhavetobelessthan1.0.</p></li></ul></li></ol><h4id="cb6ebb9e−0c50−4b08−96f3−0d6f4a32f7a2"data−toc−id="cb6ebb9e−0c50−4b08−96f3−0d6f4a32f7a2"collapsed="false"seolevelmigrated="true">TheGRASSMethodforSolvingProblems</h4><ul><li><p>Amethodusedforsolvingequationsandmakingcalculationsinphysics:</p><ul><li><p><strong>Givens</strong>:Identifyandnotethestartingvariablequantitiesprovided,includingunits.</p></li><li><p><strong>Required</strong>:Writedownthequantityyouneedtofind.</p></li><li><p><strong>Approach</strong>:Statetherelevantequationtofindtherequiredquantity.</p></li><li><p><strong>Solve</strong>:Conductthecalculations,rearrangingandsubstitutingasnecessary.</p></li><li><p><strong>State</strong>:Summarizethequantityfound,includingunits.</p></li></ul></li></ul><h5id="b93d5e8e−2911−4b59−8f81−d6e3d720b97b"data−toc−id="b93d5e8e−2911−4b59−8f81−d6e3d720b97b"collapsed="false"seolevelmigrated="true">ExampleProblem</h5><ul><li><p><strong>Problem</strong>:Determinehowfastlighttravelsthroughdiamond,giventhatitsindexofrefractionis2.42.</p><ul><li><p><strong>Givens</strong>:</p></li><li><p>.</p></li></ul></li><li><p>If a material was discovered where light traveled faster than through air/vacuum, the index of refraction would have to be less than 1.0.</p></li></ul></li></ol><h4 id="cb6ebb9e-0c50-4b08-96f3-0d6f4a32f7a2" data-toc-id="cb6ebb9e-0c50-4b08-96f3-0d6f4a32f7a2" collapsed="false" seolevelmigrated="true">The GRASS Method for Solving Problems</h4><ul><li><p>A method used for solving equations and making calculations in physics:</p><ul><li><p><strong>Givens</strong>: Identify and note the starting variable quantities provided, including units.</p></li><li><p><strong>Required</strong>: Write down the quantity you need to find.</p></li><li><p><strong>Approach</strong>: State the relevant equation to find the required quantity.</p></li><li><p><strong>Solve</strong>: Conduct the calculations, rearranging and substituting as necessary.</p></li><li><p><strong>State</strong>: Summarize the quantity found, including units.</p></li></ul></li></ul><h5 id="b93d5e8e-2911-4b59-8f81-d6e3d720b97b" data-toc-id="b93d5e8e-2911-4b59-8f81-d6e3d720b97b" collapsed="false" seolevelmigrated="true">Example Problem</h5><ul><li><p><strong>Problem</strong>: Determine how fast light travels through diamond, given that its index of refraction is 2.42.</p><ul><li><p><strong>Givens</strong>:</p></li><li><p>n = 2.42</p></li><li><p></p></li><li><p>c = 3.00 \times 10^8m/s</p></li><li><p><strong>Required</strong>:Findm/s</p></li><li><p><strong>Required</strong>: Findv(speedoflightindiamond).</p></li><li><p><strong>Approach</strong>:Usetheequation:(speed of light in diamond).</p></li><li><p><strong>Approach</strong>: Use the equation:n = \frac{c}{v}</p></li><li><p><strong>Solve</strong>:</p></li><li><p>Rearranginggives:</p></li><li><p><strong>Solve</strong>:</p></li><li><p>Rearranging gives:v = \frac{c}{n}</p></li><li><p>Substitutingvalues:<br></p></li><li><p>Substituting values:<br>v = \frac{3.00 \times 10^8 \text{ m/s}}{2.42} = 1.24 \times 10^8 \text{ m/s}</p></li><li><p><strong>State</strong>:Lighttravelsat</p></li><li><p><strong>State</strong>: Light travels at1.24 \times 10^8m/sthroughdiamond.</p></li></ul></li></ul><h5id="2f377a87−49fe−42bf−b2fd−fdf4ef9fa7e4"data−toc−id="2f377a87−49fe−42bf−b2fd−fdf4ef9fa7e4"collapsed="false"seolevelmigrated="true">FormulaTriangleforTroubleshootingAlgebra</h5><ul><li><p>Tofindonevariable:</p><ol><li><p><strong>Ifvariablesareofdifferentlevels</strong>(likem/s through diamond.</p></li></ul></li></ul><h5 id="2f377a87-49fe-42bf-b2fd-fdf4ef9fa7e4" data-toc-id="2f377a87-49fe-42bf-b2fd-fdf4ef9fa7e4" collapsed="false" seolevelmigrated="true">Formula Triangle for Troubleshooting Algebra</h5><ul><li><p>To find one variable:</p><ol><li><p><strong>If variables are of different levels</strong> (liken = \frac{c}{v}ororv = \frac{c}{n}):Dividetheothertwo.</p></li><li><p><strong>Ifvariablesareonthesamelevel</strong>:Multiplytheothertwo(forexample,): Divide the other two.</p></li><li><p><strong>If variables are on the same level</strong>: Multiply the other two (for example,c = n \times v).</p></li></ol></li></ul><h5id="3f3bf16e−7748−4c71−90c4−7e4a4671c2af"data−toc−id="3f3bf16e−7748−4c71−90c4−7e4a4671c2af"collapsed="false"seolevelmigrated="true">PracticeProblem</h5><ul><li><p><strong>Problem</strong>:Calculatetheindexofrefractionforamysterymaterialwherelighttravelsat).</p></li></ol></li></ul><h5 id="3f3bf16e-7748-4c71-90c4-7e4a4671c2af" data-toc-id="3f3bf16e-7748-4c71-90c4-7e4a4671c2af" collapsed="false" seolevelmigrated="true">Practice Problem</h5><ul><li><p><strong>Problem</strong>: Calculate the index of refraction for a mystery material where light travels at2.5 \times 10^8m/s.</p><ul><li><p><strong>Givens</strong>:</p></li><li><p>m/s.</p><ul><li><p><strong>Givens</strong>:</p></li><li><p>v = 2.5 \times 10^8m/s</p></li><li><p>m/s</p></li><li><p>c = 3.0 \times 10^8m/s</p></li><li><p><strong>Required</strong>:Findm/s</p></li><li><p><strong>Required</strong>: Findn.</p></li><li><p><strong>Approach</strong>:Use.</p></li><li><p><strong>Approach</strong>: Usen = \frac{c}{v}.</p></li><li><p><strong>Solve</strong>:<br>.</p></li><li><p><strong>Solve</strong>: <br>n = \frac{3.0 \times 10^8 \text{ m/s}}{2.5 \times 10^8 \text{ m/s}} = 1.20</p></li><li><p><strong>State</strong>:Theindexofrefractionofthemysterymaterialis1.20.</p></li></ul></li></ul><h4id="b1d6d987−95c3−483c−8507−9f9d3efd11dc"data−toc−id="b1d6d987−95c3−483c−8507−9f9d3efd11dc"collapsed="false"seolevelmigrated="true">HomeworkQuestions</h4><ol><li><p>Ifthespeedoflightinanunknownsubstanceis</p></li><li><p><strong>State</strong>: The index of refraction of the mystery material is 1.20.</p></li></ul></li></ul><h4 id="b1d6d987-95c3-483c-8507-9f9d3efd11dc" data-toc-id="b1d6d987-95c3-483c-8507-9f9d3efd11dc" collapsed="false" seolevelmigrated="true">Homework Questions</h4><ol><li><p>If the speed of light in an unknown substance is2.20 \times 10^8m/s,calculatetheindexofrefraction.(Answer:m/s, calculate the index of refraction. (Answer:n = 1.36,ethylalcohol)</p></li><li><p>Determinethespeedoflightinvegetableoilgiventhatitsindexofrefractionis1.47.(Answer:, ethyl alcohol)</p></li><li><p>Determine the speed of light in vegetable oil given that its index of refraction is 1.47. (Answer:v = 2.04 \times 10^8m/s)</p></li><li><p>Howcouldyoutellifanerrorwasmadeifthespeedoflightcalculatedinanunknownsubstancewasm/s)</p></li><li><p>How could you tell if an error was made if the speed of light calculated in an unknown substance was4.00 \times 10^8m/s?(Answer:Itisfasterthanthespeedoflightthroughavacuum,whichisimpossible.)</p></li><li><p>Aphysicsteacheroffersadiamondring,claimingitisgenuine.Aftercalculatingthespeedoflightthroughittobem/s? (Answer: It is faster than the speed of light through a vacuum, which is impossible.)</p></li><li><p>A physics teacher offers a diamond ring, claiming it is genuine. After calculating the speed of light through it to be1.56 \times 10^8m/s,wouldyouacceptthepurchase?(Answer:No,asm/s, would you accept the purchase? (Answer: No, asn = 1.92$$ which indicates it is zircon, not diamond.)

    • Explain why the index of refraction has no units. (Answer: It is a ratio of the speed of light in vacuum to the speed of light in another medium, hence units cancel out.)