Derivation of Mirror Formula: Terms, Formula, Assumptions, And Derivation

A mirror with the shape of a piece that is cut out of a spherical surface or substance is referred to as a spherical mirror or a mirror that is a part of a sphere. The concave and convex mirrors are two different varieties of spherical mirrors. A bright or dazzling spoon’s curving surface might be thought of as a curved mirror. Spherical mirrors are the most popular and often utilized kind of curved mirror. Such mirrors’ reflecting surfaces are regarded as being a component of any sphere’s surface. The term “spherical mirror” refers to mirrors with spherical reflecting surfaces.

Terms Related to Spherical Mirrors

While studying spherical mirrors, the following common phrases are important to understand:

• The capital letter C is used to indicate the curve’s center. The curve of the mirror is traversed by the point in the middle of the mirror surface, which also has the same tangent and curvature.
• The capital letter R is used to denote the radius of curvature. means that the radius of curvature is twice the focal length. Between the pole and the center of curvature, it is regarded as the linear distance.
• Principal Axis: a hypothetical line that extends from a spherical mirror’s center of curvature and through its optical center.
• Pole: The spherical mirror’s middle or center point. Capital P is used to denote it. It is the only source used for all measurements.
• A point from which the actual reflection of light occurs is called the aperture of a mirror. It also provides a sense of the mirror’s size.
• Principal Focus: The focal point is another name for the principal focus. When light rays parallel to the major axis after reflection converge, appear to converge, or appear to diverge, it is present on the axis of the mirror.
• Focus is any location on the principal axis where, after being reflected by the mirror, light rays parallel to the principal axis will converge or appear to converge.

Mirror Formula

The method employed is referred to as the mirror formula and is used to compute sums pertaining to spherical mirrors. It’s used to figure out the focal length, picture distance, object distance, magnification, and any other necessary information. In order to limit any potential error, we often compute the sums before adding the signs to the formula.

Since the sign conventions that must be adhered to when employing the mirror formula are predetermined, we can easily place the signs in the diagram above in order to obtain the desired outcome.

Typically, the object distance is regarded to be negative if the item is situated to the left of the major axis with respect to the mirror. While it is considered to be favorable if it is on the right side. The sign of the focal length depends on the type of mirror we are using; it is always positive for convex mirrors while it is negative for concave mirrors. It must be emphasized once more that in order to obtain the right answer, we must closely adhere to the sign norms.

Assumptions

There are certain assumptions related to Mirror’s Formula and these are mentioned as below:

• Measurements are made starting at the mirror’s pole for all distances.
• The positive sign denotes the distance measured in the direction of the incident ray, whereas the negative sign denotes the distance measured in the direction opposite to the incident ray, according to convention.
• While the distance above is positive, the distance above the axis is negative.

Derivation

The mirror formula’s derivation is provided below. Learners will more effectively understand the mirror formula derivation if they refer to the diagram provided below.

As per figure,

It is clear from the above illustration that the object is positioned at a distance of U from P, the mirror’s pole. We can infer from the illustration that the image is created at V by the mirror.

Now, it is evident from the accompanying figure that the opposite angles are equal in accordance with the law of vertically opposite angles. Thus, we can say:

Thus, we can say that

Now since two angles of triangle and are equal and hence the third angle is also equal and is given by;

Similarly, the triangle of and are also equal and similar, so;

Also, since is equal to so we have;

From 1 and 2 we have;

Consider that the point is very close to and hence , so;

From the above diagram and and ;

Now substituting the values of above segments along with the sign, we have;

So the above equation becomes;

Solving it we have;

since (radius of curvature is twice that of focal length), hence;

;
Solving it further and dividing with

we have;

This was how the mirror formula was derived. To gain a thorough understanding of the subject, students need to comprehend each step in the derivation of the mirror formula.

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