Understanding Image Formation in Concave Mirrors Using the Mirror Formula and Coordinate Geometry

Concave mirrors are widely used in our daily lives and scientific applications due to their ability to create highly concentrated or divergent beams of light. For instance, they are employed in lighting systems, headlights, and even in surveillance systems. In this article, we will delve into how the distance of the image from a concave mirror is calculated using the mirror formula and coordinate geometry. We will explore these concepts through real-world examples and mathematical derivations.

Using the Mirror Formula to Find the Image Distance

When dealing with concave mirrors, the mirror formula is a crucial tool for determining the image distance. It is given by:

begin{equation}frac{1}{f} frac{1}{u} frac{1}{v}end{equation}

where:

f is the focal length of the mirror. u is the object distance, taken as negative for concave mirrors. v is the image distance, which we need to find.

Example: Finding the Image Distance for a Concave Mirror

Consider a concave mirror with a radius of curvature of 20 cm. We need to find the image distance when an object is placed at a distance of 15 cm from the mirror.

Step 1: Finding the Focal Length

The focal length f of a concave mirror is calculated using the formula:

begin{equation}f -frac{R}{2}end{equation}

Given that the radius of curvature R 20 cm:

begin{equation}f -frac{20}{2} -10 text{ cm}end{equation}

Step 2: Applying the Mirror Formula

Using the mirror formula with the given values:

begin{equation}frac{1}{-10} frac{1}{-15} frac{1}{v}end{equation}

Rearranging the equation to isolate v:

begin{equation}frac{1}{v} frac{1}{-10} - frac{1}{-15}end{equation}

Finding a common denominator which is 30:

begin{equation}frac{1}{v} frac{-3}{30} - frac{2}{30} frac{-5}{30} frac{-1}{6}end{equation}

Now, taking the reciprocal to find v:

begin{equation}v -60 text{ cm}end{equation}

This means the image is located 60 cm in front of the concave mirror on the same side as the object.

Using Coordinate Geometry to Find Image Distance

Consider another example where an object is placed 25 cm from a concave mirror with a focal length of 15 cm. Let's find the image distance using coordinate geometry. The steps are as follows:

Step 1: Define the Given Values

begin{equation}f 15 text{ cm}, quad u 25 text{ cm}end{equation}

Step 2: Analyze the Ray Diagram

Assume a ray parallel to the x-axis. Let the ray BC be 1 cm above the object and the x-axis. Let alpha be the angle of inclination of the ray CE. The slope of the ray CE is:

begin{equation}m_1 tan alpha frac{1}{15}end{equation}

The x-intercept of the ray is 15 (since the ray intersects the mirror at point F, which is 15 cm from the mirror's vertex). Hence, the equation of ray CE is:

begin{equation}y frac{1}{15}x - 15end{equation}

Step 3: Apply the Law of Reflection

According to the law of reflection, the angle of incidence equals the angle of reflection. Let beta be the angle of incidence and reflection. The slope of the ray OE is:

begin{equation}m_2 tan beta frac{1}{25}end{equation}

The x-intercept of the ray is 25 (since the ray intersects the mirror at point O, which is 25 cm from the mirror's vertex). Hence, the equation of ray OE is:

begin{equation}y frac{1}{25}x - 25end{equation}

Step 4: Find the Intersection Point

To find the image, we need to find the intersection of these two lines. Setting the equations equal to each other:

begin{equation}frac{1}{15}x - 15 frac{1}{25}x - 25end{equation}

Solving for x yields the x-coordinate of the image. The y-coordinate can be found by substituting this value back into either equation. Therefore, the image is formed at a distance of 15 cm from the mirror (since y-coordinate will be 0).

Conclusion: The image distance for the second scenario is 15 cm in front of the concave mirror.