Relation Between Focal Length and Radius of Curvature

The relation between focal length and radius of curvature is one of the most important concepts in optics. 

• It describes how the shape of a spherical mirror affects the way light rays are reflected. 
• This knowledge can be used to design optical instruments, such as telescopes, microscopes, and cameras.
• A spherical mirror is a reflecting surface that is part of a hollow sphere. 
• It can be either concave or convex, depending on whether the reflecting surface is curved inwards or outwards.
• A part of a hollow sphere having an outer part coated and the inner part as a reflecting surface is called a Concave mirror.
• A part of a hollow sphere having an inner part coated and the outer part as a reflecting surface is known as a Convex mirror.
• The focal length of a spherical mirror is the distance between the pole and the principal focus.
• The radius of the sphere of which the spherical mirror forms a part is called the radius of curvature of the spherical mirror.

The relation between focal length and radius of curvature of a mirror is given by the formula

f = \(\frac{R}{2}\)

Where

• f is the focal length
• R is the radius of curvature.

Key Terms: Convex mirror, Concave mirror, Focal length, Radius of curvature, Spherical mirrors, Light, Reflecting surfaces, Plane mirror, Laws of reflection

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Spherical Mirror

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A spherical mirror is a reflecting surface that has the shape of a piece cut out of a hollow sphere. Spherical mirrors are of two types

• Concave mirror: Concave spherical mirrors are converging mirrors, which means they converge parallel beams of light into a single point. This is because the light rays are reflected off the curved surface of the mirror and towards the center of curvature.
• Convex mirror: Convex spherical mirrors are diverging mirrors, which means they diverge parallel beams of incident light. This is because the light rays are reflected off the curved surface of the mirror and away from the center of curvature.

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Focal Length and Radius of Curvature of Spherical Mirrors

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The focal length of a spherical mirror is the distance between the pole of the mirror and the principal focus. 

• The principal focus is the point on the principal axis of the mirror where all parallel rays of light incident on the mirror converge after reflection.
• It is denoted by ‘f’.
• Since focal length is a measure of distance, its SI unit is meter (m).
• The focal length of a concave mirror is taken as “Negative”.
• The focal length of a convex mirror is taken as “Positive”.
• The focal length of a plane mirror is taken as “Infinity”.

The radius of curvature of a spherical mirror is defined as the radius of the sphere of which the spherical mirror forms a part.

• It is denoted by “R”.
• Since the radius of curvature is a measure of distance, its SI unit is meter (m).
• The radius of curvature of a concave mirror is taken as “Negative”.
• The radius of curvature of a convex mirror is taken as “Positive”.
• The radius of curvature of a plane mirror is taken as “Infinity”.

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Relation Between Focal Length and Radius of Curvature Formula

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The relation between the focal length and radius of curvature of a mirror is given by the formula

f = \(\frac{R}{2}\)

Where

• f is the focal length of the spherical mirror
• R is the radius of curvature of the spherical mirror.

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Derivation of Relation Between Focal Length and Radius of Curvature of a Concave Mirror

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Let a ray OA incident on a concave mirror at point A. After reflection, the ray passes through the focus (F). CA is normal to the mirror at A.

Concave Mirror

According to the law of reflection,

∠i = ∠r = θ

Where

• ∠i is the angle of incidence
• ∠r is the angle of reflection

Also, ∠ACF = ∠OAC = θ (alternate angle)

∠AFP is the external angle of △ACF, therefore

∠AFP = ∠ACF + ∠CAF

⇒ ∠AFP = θ + θ = 2θ

From △ANC, tanθ = AN/NC

Since θ is very small, therefore tanθ ≅ θ, Hence

θ = AN/NC …(i)

From △ANF, tan2θ = AN/NF

⇒ 2θ = AN/NF …(ii)

From equation (i) and (ii), we get

2(AN/NC) = AN/NF

⇒ NC = 2 NF

As the aperture of the mirror is small, point N lies very close to P. Therefore

NF = PF and NC = PC

Thus PC = 2 PF

Using the sign convention,

• PC = – R (Radius of curvature of the concave mirror)
• PF = – f (Focal length of the concave mirror)

On substituting the values, we get

-R = – 2f

⇒ f = R/2

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Derivation of Relation Between Focal Length and Radius of Curvature of a Convex Mirror

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Let a ray of light OA incident on a convex mirror at point A, after reflection, the rays go along AB which meets the principal axis at the focus F when produced backward.

Convex Mirror

According to the law of reflection,

∠i = ∠r = θ

Where

• ∠i is the angle of incidence
• ∠r is the angle of reflection

Also, ∠CAF = ∠BAN’ = θ (vertically opposite angle)

Ans, ∠PCA = ∠OAN’ = θ (corresponding angle)

∠AFP is the external angle of △ACF, therefore

∠AFP = ∠PCA + ∠CAF

⇒ ∠AFP = θ + θ = 2θ

From △ANC, tanθ = AN/NC

Since θ is very small, therefore tanθ ≅ θ, Hence

θ = AN/NC …(i)

From △ANF, tan2θ = AN/NF

⇒ 2θ = AN/NF …(ii)

From equation (i) and (ii), we get

2(AN/NC) = AN/NF

⇒ NC = 2 NF

As the aperture of the mirror is small, point N lies very close to P. Therefore

NF = PF and NC = PC

Thus PC = 2 PF

Using the sign convention,

• PC = R (Radius of curvature of the convex mirror)
• PF = f (Focal length of the convex mirror)

On substituting the values, we get

R = 2f

⇒ f = R/2

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Solved Examples

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Ques. A concave mirror has a focal length of 20 cm. Calculate its radius of curvature.

Ans. A concave mirror is a spherical mirror in which the reflecting surface bulges inwards.

The focal length of a concave mirror is half its radius of curvature i.e.

f = R/2

Where

• f is the focal length of the concave mirror
• R is the radius of curvature of the concave mirror.

Given, the focal length of the concave mirror, f = 20 cm

Therefore, the radius of curvature, R = 2f = 2 x 20 = 40 cm

Ques. A convex mirror has a radius of curvature of 60 cm. Calculate its focal length.

Ans. A convex mirror is a spherical mirror in which the reflecting surface bulges outward.

The radius of curvature of the convex mirror is related to its focal length as

f = R/2

Where

• f is the focal length of the convex mirror
• R is the radius of curvature of the convex mirror.

Given, the radius of curvature of the convex mirror, R = 60 cm

Therefore, the focal length, f = R/2 = 60/2 = 30 cm

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Things to Remember

• A spherical mirror is a reflecting surface that is formed by a part of a hollow sphere.
• The radius of curvature of a spherical mirror is the distance between the pole of the mirror and the center of curvature.
• The distance between the pole and the principal focus of the spherical mirror is called the focal length of the spherical mirror.
• The relationship between the focal length and radius of curvature of a spherical mirror is given by f = R/2.
• The relation f = R/2 is valid for both concave and convex mirrors.
• The focal length and radius of curvature of a plane mirror is infinity.
• The focal length of a mirror does not change when immersed in a liquid.

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