# Determination of Focal Length of Concave Mirror and Convex Mirror

## What are Convex and Concave Mirrors?

A curved mirror in which a reflective surface bulges out towards the light source is known as convex mirror. The convex mirror reflects the light outwards and so it is not used to focus light. As the object comes nearer to the mirror, the size of the object gets larger until it reaches its original size. These mirrors are also known as the diverging mirrors.

A concave mirror has a reflecting surface that caves inwards. The mirror also converges the light at one prime focus point; hence they are also called as converging mirrors. They are applied to focus light. Depending upon the location of the object with respect to the mirror, the size of the image formed by the concave mirror varies. It can be real or virtual, inverted or erect and magnified, reduced, or be similar in size of the object depending upon the position.

### Focal Length of Concave Mirror

This article will help you find the focal length of a concave mirror. Let’s look at the theory to obtain the image of a farther object.

1. Like a plane mirror, the concave mirror obeys the law of reflection of light.

2. Ray of light from an object – The rays of light emitted from a distant object, e.g., distant buildings or sun, are parallel to each other. When the parallel rays from the source fall on the concave mirror along the axis, reflect and meet at the point in front of the mirror, which is known as the mirror's principal focus.

3. At the focus of the mirror, a real, inverted, and very small image size is formed.

1. Focal length – Focal length of the concave mirror is the distance between the pole P of concave mirror and the focus F. By obtaining the Real image of the distant object, the focal length of concave mirror can be determined, as shown in the diagram.

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### Focal Length of Concave Mirror Formula

Lets see the above shown diagram,

Focal Length – The space between the pole P of concave mirror and the focus F is the focal length of concave mirror. By obtaining the Real image of a distant object at its focus, the focal length of the concave mirror can be estimated as shown in the diagram.

The focal length of the convex mirror is positive, whereas that of the concave mirror is negative. The same can also be proved by using the mirror formula:

(1/f = 1/v +1/u).

Let's see how

Since we know that an object is always placed at the left side or direction opposite the incidence ray of the mirror, the object distance will always be negative.

u = -u

v = -v (Image distance is negative since image produced by concave mirrors are usually on the left side or direction opposite to the incidence ray)

Using mirror formula,

1/f = 1/v +1/u

Or, 1/f = (-u-v)/uv

Or, f = -uv/(u+v)

### Focal Length of Convex Mirror Using Convex Lens

A curved mirror in which mirroring surface bulges towards the light source is known as the convex mirror. The light is reflected outwards in a convex mirror; therefore, they are not used to focus light. The convex mirror is also called a diverging mirror or fish-eye mirror.

The image created by a convex lens is erect and virtual since the focal point (F), and center of curvature (2F) are both imaginary points within the mirror that cannot be reached. As a result, the image formed by these mirrors cannot be projected on the screen as the image is inside the mirror. Hence the focal length cannot be determined directly. Initially, the size of the image is smaller than the object, but it gets larger as the object approaches the mirror. The diagram below shows the convex mirror.

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The focal length of a convex mirror can be determined by introducing the convex lens between the object and the convex mirror. With the help of a convex lens side by side with an object, an image can be obtained when the convex mirror reflects the rays along the same path, i.e. when rays fall naturally on the mirror. The space between the screen and the mirror is the radius of curvature, which is denoted by R.

By using the formula below, the focal length f of the convex mirror can be calculated.

F=(Radius of Curvature R)/2

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FAQ (Frequently Asked Questions)

1. What is a Relation Between Focal Length and Radius of Curvature.

Ans- The relationship between the radius of curvature and focal length r is represented by r =2f

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Consider a ray of light AB which is parallel to the principal axis and incident on a spherical mirror at point B. The normal to surface at point B is CB, and CP = CB = R is the radius of curvature. The ray AB following reflection from the mirror will pass via F(concave mirror) or will appear to diverge from F(convex mirror) and obeys the law of reflection i.e., i = r.

From the geometry of the diagram,

∠BCP = θ = i

In D CBF, θ = r

∴BF = FC (because i = r)

If the hole of the mirror is small, B lies close to P, and therefore BF = PF.

=>FC = FP = PF

=>PC = PF + FC = PF + PF

=>R = 2 PF = 2f

=>f = R/2

Similar relation stands for convex mirror also. In obtaining this relation, we have believed that the aperture of the mirror is small.

2. Why is a Driving Mirror of Vehicles made of a Convex Mirror?

Ans-For all positions of an object, a convex mirror forms virtual, erect, and diminished image. As the image diminished in size, a broader field of view behind the vehicle is covered. The two qualities of the image formed by the convex mirror, viz. a wider and erect field of sight helps the driver in driving the vehicle with ease.

3. What is the Focal Length of a Concave Lens and Convex Lens?

Ans -

Convex Lens

The focal length is positive

Image obtained can be either virtual or real

The light rays get converged towards the principal axis

Concave Lens

The focal length is negative

Image obtained is always virtual

The light ray diverges away from the principal axis