Answer
Hint:The radius of curvature of convex or concave mirror is equal to two times of the focal length of convex or concave mirror.The radius of curvature is the radius of sphere formed by the convex or concave mirror. It is also equal to the distance between the pole and centre of curvature. The sign convention for focal length and radius of curvature is the same.
Complete step by step answer:
From the question, we know that the radius of curvature of a convex mirror is, $R = 40\;{\rm{cm}}$We know that the relationship between the radius of curvature of convex mirror and its focal length,$f = \dfrac{R}{2}$Here, $f$ is the focal length of the convex mirror.Substitute the given value in the above equation, we have,$f = \dfrac{{40\;{\rm{cm}}}}{2}\\\therefore f = 20\;{\rm{cm}}$Thus, the focal length of the convex mirror is 20 cm and option C is correct.
Note:In convex mirror, the mirror coating is inside the spherical surface. The image formed by the convex mirror is smaller than the object; it will be upright and virtual. The image will be closer to the mirror as compared to the object whereas in concave mirror, the mirror coating is outside the spherical surface. The image formed by the concave mirror is bigger than the object; it will be upright and virtual. The image will be behind the mirror.
Option 2 : 60 cm from the mirror on the side of the object
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Concept:
- Spherical mirrors are of two types - Convex mirror, and Concave mirror.
- The image formed by the spherical mirrors depends on the position of the object.
- The relationship between the focal length (f) and the radius of curvature (R) is given by -
\(f = \frac{R}{2}\)
- Concave mirror: If the inner surface of the spherical mirror is the reflecting surface, then it is called a concave mirror. It is also called a focusing mirror/converging mirror.
- The size of the image produced by these mirrors can be larger or smaller than the object, depending upon the distance of the object from the mirror.
- The concave mirror can form both real as well as virtual images of any object.
- Mirror formula: The expression which shows the relation between object distance (u), image distance (v), and focal length (f) is called mirror formula.
\(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\)
Calculation:
Given,
Radius of curvature of concave mirror (R) = -40 cm
So focus = R/2 = -40/2 = -20 cm
Object is placed in front of mirror (u) = -30 cm
Image position (v) =?
We know that,
\(\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \Rightarrow \frac{1}{v} = \frac{1}{f} - \frac{1}{u}\)
\( \Rightarrow \frac{1}{v} = \frac{1}{{ - 20}} - \frac{1}{{ - 30}} = \frac{{ - 3 + 2}}{{60}} = - \frac{1}{{60}}\)
\(\frac{1}{v} = - \frac{1}{{60}} \Rightarrow v = - 60\;cm\)
The image is formed at 60 cm in front of mirror or on the side of object.
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In optics, we have come across different types of mirrors and have studied their properties in detail like the radius of curvature, focal length, focal point, dimension of the mirror, imaging capacity (erect or inverted), thickness, refractive index and many more. Mirrors are divided into two types:
- Plane Mirrors
- Spherical Mirrors
In this article, let us understand the relationship between focal length and radius of curvature.
The reflecting surface of a spherical mirror may be curved inwards or outwards. The focal length of a mirror is represented as f and is defined as the distance between the focus and the pole of the mirror. The radius of curvature is represented as R and is defined as the radius of the mirror that forms a complete sphere.
A ray of light AB, which is incident on a spherical mirror at point B and is parallel to the principal axis. CB is normal to the surface at point B. CP = CB = R is the radius of curvature. After reflection from mirror the light will pass through the focus of the concave mirror F or will appear to diverge from the focus of the convex mirror F and obeys the law of reflection i.e. i = r.
From the geometry of the figure,
∠BCP = θ = i (As ∠BCP and ∠ABC are alternate angles)
In ΔCBF, ∠CBF = θ = r
∴BF = FC (because i = r)
If the aperture of the mirror is small, B lies close to P, and therefore BF = PF
Or FC = BF = PF
Or PC = PF + FC = PF + PF
Or R = 2 PF = 2f
Or f = R/2
This kind of relation can be applied for convex mirrors too. In this relation, the aperture of the mirror is assumed to be small.
Radius of curvature is observed to be equal to twice the focal length for spherical mirrors with small apertures. Hence R = 2f .
We can say clearly that the principal focus of a spherical mirror lies at the centre between the centre of curvature and the pole.
Read more about Electricity and magnetism.
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A mirror is a reflective surface that bounces off light, producing either a real image or a virtual image.
- Plane Mirrors
- Spherical Mirrors
Spherical mirrors are of two types as:
- Concave Mirror
- Convex Mirror
A convex mirror is curved in shape where the reflective surface bulges out towards the light source.
The centre of the curvature (C).