Explain how to determine the wavelength of light using a laser and a diffraction grating, including a labeled…

Question

Explain how to determine the wavelength of light using a laser and a diffraction grating, including a labeled diagram.

ScanSolve AI Tutor · Accepted Answer

Here's how to determine the wavelength of light using a laser and a diffraction grating:

Diagram (2 marks)

\text{
    \begin{tikzpicture}[
        thick,
        laser/.pic={
            \draw (0,0) rectangle (1,0.5);
            \draw[fill=red] (0.8,0.25) circle (0.1);
            \draw[red,->] (1,0.25) -- (2,0.25);
            \node at (0.5,0.7) {Laser};
        },
        grating/.pic={
            \draw (0,0) -- (0,1);
            \foreach \y in {0.1,0.2,...,0.9} \draw (0,\y) -- (0.2,\y);
            \node at (0.1,1.2) {Grating};
        },
        screen/.pic={
            \draw (0,0) rectangle (0.2,3);
            \node at (0.1,3.2) {Screen};
        }
    ]
        % Laser
        \pic at (0,1.5) {laser};

        % Diffraction Grating
        \pic at (3,1.5) {grating};

        % Screen
        \pic at (8,0) {screen};

        % Laser beam path
        \draw[red,->] (2,1.75) -- (3,1.75); % From laser to grating

        % Diffracted beams
        \draw[red,->] (3.2,1.75) -- (8,1.75); % Central maximum (n=0)
        \draw[red,->] (3.2,1.75) -- (8,2.5); % First order (n=1)
        \draw[red,->] (3.2,1.75) -- (8,1.0); % First order (n=1)

        % Labels for maxima
        \node at (8.5,1.75) {$n=0$};
        \node at (8.5,2.5) {$n=1$};
        \node at (8.5,1.0) {$n=1$};

        % Distances
        \draw[<->] (3.1,0.5) -- (8.1,0.5);
        \node at (5.6,0.7) {L}; % Distance from grating to screen

        \draw[<->] (8.3,1.75) -- (8.3,2.5);
        \node at (8.6,2.125) {$x$}; % Distance from central to first order

        % Angle
        \draw (3.2,1.75) -- (8,1.75);
        \draw (3.2,1.75) -- (8,2.5);
        \pic[draw, angle radius=1cm, "$\theta$"] {angle = 8-1.75-3.2 -- 8-1.75-8 -- 8-2.5-8};
    \end{tikzpicture}
}

Observation (2 marks)

When the laser beam passes through the diffraction grating, a pattern of bright spots (maxima) is observed on the screen.
There is a very bright central spot (the central maximum, $n=0$ ).
Symmetrically on either side of the central maximum, there are other bright spots (first-order maxima, $n=1$ ; second-order maxima, $n=2$ , etc.).
The intensity of the bright spots decreases as the order ( $n$ ) increases, moving away from the central maximum.

Calculations (4 marks)

Determine grating spacing ( $d$ )
The diffraction grating is usually labeled with the number of lines per millimeter (e.g., 300 lines/mm). Convert this to lines per meter, then find the spacing $d$ between adjacent lines:
$d = \frac{1}{\text{number of lines per meter}}$
For example, if it's 300 lines/mm:
$d = \frac{1}{300 \text{ lines/mm}} = \frac{1}{300 \times 1000 \text{ lines/m}} = \frac{1}{3 \times 10^5 \text{ lines/m}}$ $d = 3.33 \times 10^{-6} \text{ m}$
Measure distances
- Measure the distance $L$ from the diffraction grating to the screen.
- Measure the distance $2x$ between the two first-order maxima (the bright spots on either side of the central maximum). Then, $x = \frac{2x}{2}$ .
Calculate the angle of diffraction ( $\theta$ )
Use trigonometry:
$\tan \theta = \frac{x}{L}$
Then, calculate $\theta = \arctan\left(\frac{x}{L}\right)$ .
For small angles, $\sin \theta \approx \tan \theta$ .
Calculate the wavelength ( $\lambda$ )
Use the diffraction grating formula:
$d \sin \theta = n \lambda$
For the first-order maximum, $n=1$ . Rearrange to find $\lambda$ :
$\lambda = \frac{d \sin \theta}{n}$
Substitute the values of $d$ , $\sin \theta$ , and $n=1$ to obtain the wavelength of the laser light.

Precaution (2 marks)
- Laser Safety: Never look directly into the laser beam as it can cause permanent eye damage. Ensure the beam is directed away from people's eyes.
- Dark Room: Conduct the experiment in a darkened room to clearly observe the diffraction pattern and accurately measure the positions of the maxima.
- Perpendicular Alignment: Ensure the laser beam is perpendicular to the diffraction grating to avoid errors in the angle measurement.
- Accurate Measurements: Take multiple readings for $x$ and $L$ and calculate their averages to minimize random errors.

Explain how to determine the wavelength of light using a laser and a diffraction grating, including a labeled diagram.

Step-by-step solution

Determine grating spacing ( $d$ )

Measure distances

Calculate the angle of diffraction ( $\theta$ )

Calculate the wavelength ( $\lambda$ )

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More Physics Questions

Explain how to determine the wavelength of light using a laser and a diffraction grating, including a labeled diagram.

Step-by-step solution

Determine grating spacing (ddd)

Measure distances

Calculate the angle of diffraction (θ\thetaθ)

Calculate the wavelength (λ\lambdaλ)

Still stuck on this one?

More Physics Questions

Determine grating spacing ( $d$ )

Calculate the angle of diffraction ( $\theta$ )

Calculate the wavelength ( $\lambda$ )