
Derive the relation between Focal length and radius of curvature for spherical mirrors.
Answer
522.9k+ views
Hint: Focal length and radius of curvature directly depends on each other. Focal length and radius of curvature for a mirror have the same unit. Also here we need to apply the rules or laws of reflection.
Complete step-by-step answer:
To find relation between focal length and radius of curvature for spherical mirror
Consider a concave mirror
Where,
P = Pole, it is the midpoint of mirror
F = Focal point
C = center of curvature
Cp = Radius of creature
Ray parallel to principal axis strike at point A.
For Applying rules of reflection we have to draw a normal
As we know the line which strikes circular. Also we can say normal at the surface of the sphere will pass through the Center of curvature of mirror or spherical shapes.
Now According to Rules Angle of incident = Angle of Reflection
\[\angle i=\angle r\]
AS OA and OP are parallel to each other
\[\angle \text{ACE}=\text{ }i\text{ }\left( \text{Alternate Angle} \right)\text{ }..........\text{ 1}\]
Also \[\angle \text{FAC}=i\text{ }\left( \text{Reflection Angle} \right)~~~\text{ }\ldots \ldots \ldots \text{ }2\].
By equation 1 and 2
\[\angle \text{ACF}=\angle \text{FAS}=i\]
Therefore CF = FA
When incident way come closer to principal axis then FA= FP
Also we know CF = FP = f (focal length)
Then \[R=CP\]
\[=CF+FP\]
\[=F+F\]
\[=2F\]
So, Radius of curvature is double the focal length.
Note: The Principal focus of a spherical mirror lies in between the role and center of curvature. Also, the Radius of curvature is equal to twice of the focal length.
In general as we know we cut mirrors from the sphere. So the radius of that sphere from which the mirror has cut is known as the radius of curvature of the mirror. There is a focal point for every mirror. So focal length is defined as distance between focal point and mirror.
Complete step-by-step answer:
To find relation between focal length and radius of curvature for spherical mirror
Consider a concave mirror

Where,
P = Pole, it is the midpoint of mirror
F = Focal point
C = center of curvature
Cp = Radius of creature
Ray parallel to principal axis strike at point A.
For Applying rules of reflection we have to draw a normal
As we know the line which strikes circular. Also we can say normal at the surface of the sphere will pass through the Center of curvature of mirror or spherical shapes.
Now According to Rules Angle of incident = Angle of Reflection
\[\angle i=\angle r\]
AS OA and OP are parallel to each other
\[\angle \text{ACE}=\text{ }i\text{ }\left( \text{Alternate Angle} \right)\text{ }..........\text{ 1}\]
Also \[\angle \text{FAC}=i\text{ }\left( \text{Reflection Angle} \right)~~~\text{ }\ldots \ldots \ldots \text{ }2\].
By equation 1 and 2
\[\angle \text{ACF}=\angle \text{FAS}=i\]
Therefore CF = FA
When incident way come closer to principal axis then FA= FP
Also we know CF = FP = f (focal length)
Then \[R=CP\]
\[=CF+FP\]
\[=F+F\]
\[=2F\]
So, Radius of curvature is double the focal length.
Note: The Principal focus of a spherical mirror lies in between the role and center of curvature. Also, the Radius of curvature is equal to twice of the focal length.
In general as we know we cut mirrors from the sphere. So the radius of that sphere from which the mirror has cut is known as the radius of curvature of the mirror. There is a focal point for every mirror. So focal length is defined as distance between focal point and mirror.
Recently Updated Pages
Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

While covering a distance of 30km Ajeet takes 2 ho-class-11-maths-CBSE

Sanjeevani booti brought about by Lord Hanuman to cure class 11 biology CBSE

A police jeep on patrol duty on a national highway class 11 physics CBSE

Master Class 11 Economics: Engaging Questions & Answers for Success

Trending doubts
Which one is a true fish A Jellyfish B Starfish C Dogfish class 10 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

When and how did Canada eventually gain its independence class 10 social science CBSE

What is the median of the first 10 natural numbers class 10 maths CBSE
