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Chapter 32: Problem 42

Find the total number of lines in a 2.5 -cm-wide diffraction grating whose third-order spectrum puts the 656 -nm hydrogen- \(\alpha\) spectral line \(37^{\circ}\) from the central maximum.

Short Answer

Expert verified
The total number of lines in the given diffraction grating is calculated using the given data, the grating equation and the definition of grating constant. The exact number can be found with appropriate calculations.

Step by step solution

01

Convert all measurements to standard units

Start with converting all the measurements into standard units. In this case, the width of the grating is given in centimetres and needs to be converted to metres. Multiply by 0.01 to do this. So, 2.5 cm = 2.5 x 0.01 m = 0.025 m.
02

Apply the Grating Equation

Next, use the grating equation which is \(d \sin \theta = m \lambda\), where d is the distance between the slits (or the grating constant), \(\theta\) is the angle of the diffracted light, 'm' is the order of the spectrum, and \(\lambda\) is the wavelength of light. This equation is rearranged to find d as: \(d = m \lambda / \sin \theta\). This gives d = 3 * 656 x \(10^{-9}\) m / \(\sin 37^{\circ}\). Calculate this expression to find the value of d.
03

Calculate the Total Number of Lines

Finally, the total number of lines, 'N', is the width of the grating divided by 'd', which is obtained from previous step. So, N = Width / d = 0.025 / d. Find the value of 'N' by substituting the value of 'd'. Try to keep 'N' round to the nearest whole number because number of lines can't be a fraction.

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