Problem 2 - College Physics

Question

In the development of Young's double-slit experiment, a small-angle approximation \((\tan \theta \approx \sin \theta)\) was used to find the lateral displacements of the maxima (bright) and minima (dark) positions. How good is this approximation? For example, what is the percentage error for \(\theta=10^{\circ} ?\)

Step-by-Step Solution

Answer

The percentage error for \( \theta = 10^{\circ} \) is approximately 1.55\%.

Step 1: Understand the small-angle approximation

The small-angle approximation states that for small angles, typically measured in radians, the values of \( \tan(\theta) \) and \( \sin(\theta) \) are approximately equal. This is often used when \( \theta \) is small (usually below 15°) to simplify calculations in trigonometry.

Step 2: Convert degrees to radians

Angles in trigonometric calculations like this should be in radians. Convert \( \theta = 10^{\circ} \) to radians using the conversion \( 1^{\circ} = \frac{\pi}{180} \, \text{radians} \). Thus, \( 10^{\circ} = \frac{10 \pi}{180} = \frac{\pi}{18} \approx 0.1745 \, \text{radians} \).

Step 3: Calculate \( \tan(\theta) \) and \( \sin(\theta) \)

Evaluate \( \tan(0.1745) \) and \( \sin(0.1745) \) using a calculator. For \( \theta = 0.1745 \): \( \tan(0.1745) \approx 0.1763 \), \( \sin(0.1745) \approx 0.1736 \).

Step 4: Determine the difference

Find the difference between \( \tan(\theta) \) and \( \sin(\theta) \) by subtracting the two: \( \tan(0.1745) - \sin(0.1745) = 0.1763 - 0.1736 = 0.0027 \).

Step 5: Percentage error calculation

To find the percentage error, use the formula: \( \text{Percentage Error} = \left( \frac{|\tan(\theta) - \sin(\theta)|}{\sin(\theta)} \right) \times 100 \). So, \( \text{Percentage Error} = \left( \frac{0.0027}{0.1736} \right) \times 100 \approx 1.55\%. \)

Key Concepts

Small-Angle Approximation

In physics, the small-angle approximation is a powerful simplification, especially in scenarios where the angle \(\theta\) is small enough that \(\tan \theta\) and \(\sin \theta\) can be treated as nearly equal. This approximation often applies when \(\theta\) is less than about 15 degrees. The simplification allows physicists and engineers to make complex calculations more manageable.

• The basic principle centers around measuring angles in radians. In radians, small angles appear very close in value when comparing \(\tan \theta\), \(\sin \theta\), and even \(\theta\) itself.


• For instance, in Young's double-slit experiment, this approximation helps in predicting points of constructive and destructive interference without complicating trigonometric calculations.

Understanding how these functions behave for small angles is essential. For instance, if \(\theta = 10^{\circ}\), translating this into radians involves using the conversion \(1^{\circ} = \frac{\pi}{180}\) radians, yielding \(0.1745\) radians. At this value, the difference between \(\tan(0.1745)\) and \(\sin(0.1745)\) is minimal, making this approximation useful.

Trigonometry in Physics

Trigonometry is a cornerstone of physics, enabling the study of angles and their relationships in different scenarios, such as in Young's double-slit experiment. There, trigonometry helps analyze the path differences and interference patterns arising when two waves meet. It plays a crucial role in understanding various physical phenomena.

• Trigonometric functions like sine and tangent describe the ratios of sides in a right triangle relative to an angle.


• In practical physics applications, these functions assist in outlining the angles and dimensions of physical scenarios involving waves, projectiles, and optics.

By using the small-angle approximation, the trigonometric complexity reduces, allowing for simpler computations with nearly the same accuracy. For \(\theta = 10^{\circ}\) (or \(0.1745\) radians), the calculations for \(\tan(\theta)\) and \(\sin(\theta)\) produce similar values. This makes the approximation very useful in simplifying studies of light behavior and mechanics of motion.

Percentage Error Calculation

Calculating the percentage error is a fundamental skill in mathematics and physics, providing a measure of an approximation's accuracy. In the context of the small-angle approximation, it quantifies how close \(\tan(\theta)\) is to \(\sin(\theta)\), stipulating the precision of using one in the place of the other.

• The formula for percentage error is \(\text{Percentage Error} = \left( \frac{|\tan(\theta) - \sin(\theta)|}{\sin(\theta)} \right) \times 100\).


• It expresses the error as a percentage of the actual value, helping to assess the reliability of approximations in calculations.

For example, when \(\theta = 10^{\circ}\), the small difference \(0.0027\) between these trigonometric values results in a percentage error of approximately \(1.55\%\). This low error demonstrates the effectiveness of the small-angle approximation within its limitations, providing confidence in its usage for simplifying trigonometric problems in physics.