Register & enjoy the perks of a Universal College Application, Easy Process, Time-saving, and Free Contact details and Brochures.
Monthly Users
Monthly Applications
Admissions
No Result Found
Home > News & Articles > Mirror Formula Derivation: Definition, Formula Derivation (1/f = 1/v + 1/u), Equation, Applications and Proof
Updated on 02nd January, 2024 , 8 min read
The mirror formula, also known as the mirror equation, is obtained through a derivation that establishes a relationship between the object distance, image distance, and focal length. It is applicable to both plane mirrors, which produce straight images that reverse the front and back of real objects, and spherical mirrors, such as concave and convex mirrors. Spherical mirrors have a shape resembling a piece cut from a spherical surface or material. The derivation of the mirror formula is a commonly encountered topic in board and competitive examinations. In this article, we will explore the derivation process and provide sample questions related to the mirror formula.
Spherical mirrors are curved glass surfaces that reflect light back to the surface. By understanding the properties of these mirrors, we can derive the formulas for calculating their magnification and other properties.
Let's start with basic anatomy: Spherical mirrors have two parts—the radius of curvature (R) and the focal point (F). Radius of curvature (R) is the distance from the mirror's center to its edge, while focal point (F) is the point where all incoming light converges when it reflects off the mirror.
The shape of a spherical mirror can vary depending on its curvature. Mirrors with short radii of curvature (R1) will produce a convex mirror, while mirrors with long radii of curvature (R2) will produce a concave mirror. In either case, the reflected images will be distorted depending on how far away they are from the focal point (F).
By understanding these basic properties of spherical mirrors, we can begin to derive the formulas for calculating their magnification and other characteristics. This is essential for professionals in all fields who depend on accurate readings from mirrors, from optics engineers to physicists.
The mirror formula is an equation that relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror. It is commonly used in optics to determine the position and characteristics of an image formed by a mirror.
The mirror formula is expressed as:
1/f = 1/v + 1/u
Where:
This formula is derived from the lens/mirror maker's formula, which is based on the principles of refraction and reflection. By using the mirror formula, one can calculate the focal length of a mirror if the object and image distances are known, or vice versa. It is a fundamental tool in understanding the behavior of spherical mirrors in optics.
In the derivation of the mirror formula, a sign convention is followed to determine the sign (+/-) of the object distance (u), image distance (v), and focal length (f). This convention helps in representing the direction and nature of the distances and focal length involved in the formula. The sign convention for mirror formula derivation is as follows:
By adhering to this sign convention, the mirror formula equation (1/f = 1/v + 1/u) can be used to correctly determine the characteristics and position of the image formed by the mirror.
The mirror formula is derived based on a few assumptions. These assumptions include:
Consider the ray diagram where the object AB is positioned on the principal axis of a concave mirror. The object is placed beyond the center of curvature (C), resulting in the formation of an image A'B' between the center of curvature (C) and the principal focus (F) of the concave mirror. In this scenario, we are specifically considering a concave mirror that forms a real image.
By examining the ray diagram, we can observe that triangles ΔABC and ΔA′B′C are similar. Therefore, the ratios of their corresponding sides are equal:
AB / A′B′ = CB / CB′
Similarly, triangles ΔABP and ΔA′B′P are also similar, leading to the relationship:
AB / A′B′ = PB / PB′
Combining these two relations, we find:
AB / A′B′ = CB / CB′ = PB / PB′
Next, measuring distances from the pole (P), we have:
CB = PB – PC
CB′ = PC – PB′
Substituting the values of CB and CB' into the equation above, we get:
(PB – PC) / (PC – PB′) = PB / PB′
Applying the Cartesian sign convention, we assign the following values:
PB = –u
PC = –R
PB′ = –v
Replacing PB, PB', and PC in the equation, we have:
(-u - (-R)) / (-R - (-v)) = -u / -v
Simplifying further, we obtain:
(-u + R) / (-R + v) = u / v
Further simplification leads to:
v(-u + R) = u(-R + v) - vu + vR
= -uR + uv
Alternatively, we can express it as:
uR + vR = 2uv
Dividing both sides by uvR, we get:
1/v + 1/u = 2/R
Since we know that f = R/2, we can substitute it into the equation:
1/f = 1/v + 1/u
This equation holds true even when a virtual image is formed by a concave mirror.
Hence, the mirror equation for a spherical mirror is:
1/f = 1/v + 1/u
For a convex lens, the mirror formula is:
1/f = 1/v - 1/u
The "Mirror Formula" is a mathematical expression of how light rays are reflected off of a curved mirror. It is composed of two variables: object distance (do) and image distance (di).
Object Distance
Object distance is the distance between the object and the center of curvature of the reflecting surface. This distance is measured in meters, with a negative sign indicating that the object is located on the same side as the mirror.
Image Distance
Image distance is determined by measuring how far away from the center of curvature the reflected image appears. This distance is also measured in meters, with a positive sign indicating that it appears on the opposite side from the mirror.
Together, these two variables provide an accurate reflection of light rays - that is why they are used to determine image distances for various applications such as lens design and beam divergence calculations. By taking into account both object and image distances, you can ensure accurate reflections from your mirrors.
The mirror formula describes how light reflects off of a curved surface, such as a convex or concave mirror. For instance, if you were to draw a line from the center of curvature (C) to the focal point (F) and then another line from the center of curvature to any point on the mirror's surface (Q), then the angle between these two lines will be equal to twice the angle between F and Q. Once this angle is known, calculating the reflected light follows from basic trigonometric principles.
The same principle applies for acoustic waves. If one were to place a speaker at C and a microphone at Q, then measuring the angle between these two points will tell you exactly how sound will reflect off of the curved surface. From this data, it's possible to determine how loud certain frequencies would be in different parts of a room - allowing us to better tune our acoustics for optimal sound quality.
Finally, optical techniques such as spectroscopy depend heavily on understanding how electromagnetic radiation interacts with different materials. By using our mirror formula, we can calculate exactly where radiation is most likely to reflect off of an object - allowing us to make precise measurements about its composition.
The Mirror Equation can be utilized in the following manners:
In conclusion, the process of mirror formula derivation is a crucial mathematical tool that can be used to solve problems related to ray optics and to find the characteristics of images produced in mirrors. Understanding the basic steps and formulas associated with the mirror formula derivation is a fundamental requirement for anyone dealing with optics, allowing quick and accurate calculations that provide useful results. Through these calculations, the properties of images formed in mirrors can be accurately determined. So, the next time you're looking for the image characteristics in a mirror, remember the process of mirror formula derivation and use it to maximize your accuracy and efficiency.
Was this Article Helpful/Relevant or did you get what you were looking for ?
👍 1,234
👎234
D Pharmacy: Admission 2024, Subjects, Colleges, Eligibility, Fees, Jobs, Salary
By - Nikita Parmar 2023-11-04 08:11:48 , 7 min readHow to Become a Pilot after 12th: In India, Courses, Exams, Fees, Salary 2023
By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readThe mirror formula derivation is a mathematical process that establishes the relationship between the object distance (u), image distance (v), and focal length (f) of a mirror. It allows us to predict and calculate the properties of images formed by mirrors.
The mirror formula derivation assumes a thin mirror, small angles, paraxial rays, spherical mirror shape, and adherence to the laws of reflection.
The derivation involves constructing similar triangles using the ray diagram and applying the ratios of corresponding sides. This leads to the equation connecting object distance, image distance, and focal length.
Yes, the mirror formula derivation is applicable to both concave and convex mirrors. However, specific examples may focus more on concave mirrors.
The Cartesian sign convention helps assign positive or negative values to object distance (u), image distance (v), and focal length (f) based on their relative positions with respect to the mirror. It ensures consistency in calculations.
The mirror formula derivation is valid for both real and virtual images formed by concave mirrors. It provides accurate predictions of image properties in both cases.
The small-angle approximation assumes that the angles involved in the derivation are small, allowing for simplified calculations and a more manageable derivation process.
The mirror formula derivation is specifically based on the assumption of a spherical mirror shape. While it may not hold precisely for non-spherical mirrors, it still provides a useful approximation in many practical scenarios.
The mirror formula derivation and the magnification equation are complementary. When one is known, the other can be derived using the mirror formula and the concept of magnification.
The mirror formula derivation enables you to calculate and predict image characteristics, such as size, position, and orientation, for different mirror configurations. It is widely used in optics, engineering, and various scientific applications.
communication@collegesearch.in
+91 84482 99862 
Download Brochure
