9.2-Spherical Mirrors

9.2-Spherical Mirrors Important Formulae

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Grade 10 → Science → Light-Reflection and Refraction → 9.2-Spherical Mirrors

Spherical mirrors are mirrors that have the shape of a section of a sphere. They are classified into two main types: concave mirrors and convex mirrors. Each type has unique properties and applications based on the way they reflect light.

Concave Mirrors

Concave mirrors are curved inward, resembling a bowl. They have several important features:

  • Principal Axis: The principal axis is an imaginary line that passes through the center of the mirror and its focal point.
  • Focal Point (F): The point where parallel rays of light converge after reflecting off the mirror. The focal length (f) is the distance from the mirror's surface to the focal point.
  • Center of Curvature (C): The center of the sphere from which the mirror is derived. It is located at a distance of twice the focal length (C = 2f) from the mirror.
Properties of Concave Mirrors

Concave mirrors have specific image formation properties:

  • When the object is placed beyond the center of curvature (C), the image formed is real, inverted, and smaller than the object.
  • When the object is at the center of curvature (C), the image is real, inverted, and of the same size as the object.
  • When the object is between the center of curvature (C) and the focal point (F), the image is real, inverted, and larger than the object.
  • When the object is at the focal point (F), no image is formed since the rays of light will reflect parallel to the principal axis.
  • When the object is between the focal point (F) and the mirror, the image formed is virtual, upright, and larger than the object.
Applications of Concave Mirrors

Concave mirrors are widely used in various applications, including:

  • Shaving Mirrors: They provide a magnified and upright image, making it easier to see while shaving.
  • Telescope Mirrors: Concave mirrors are used in reflecting telescopes to gather and focus light from distant stars and celestial bodies.
  • Satellite Dishes: They reflect signals to a focal point where the receiver is located, enhancing signal strength.
Convex Mirrors

Convex mirrors are curved outward and have distinct properties:

  • The principal axis is also present, extending through the mirror's surface.
  • Convex mirrors have a virtual focal point located behind the mirror where light rays appear to diverge from.
  • As the focal point is virtual, the focal length (f) is considered negative.
Properties of Convex Mirrors

Convex mirrors form images with specific characteristics:

  • Regardless of the object's position, the image formed by a convex mirror is always virtual, upright, and smaller than the object.
  • This is because the light rays diverge after reflecting off the mirror, making it appear as if they originate from a point behind the mirror.
Applications of Convex Mirrors

Convex mirrors have practical applications in various fields:

  • Vehicle Rear-View Mirrors: They provide a wider field of view, allowing drivers to see more of the area behind them.
  • Security Mirrors: Used in stores and parking lots to monitor areas and prevent theft.
  • Road Safety: Convex mirrors are installed at intersections and blind curves to enhance visibility and prevent accidents.
Image Formation by Spherical Mirrors

The formation of images by spherical mirrors can be understood through ray diagrams, which illustrate the paths of light rays as they reflect off the mirror surfaces. Key rays include:

  • Ray parallel to Principal Axis: After reflection, this ray passes through the focal point (F) in concave mirrors and appears to diverge from F in convex mirrors.
  • Ray through F: In concave mirrors, this ray reflects parallel to the principal axis, while in convex mirrors, it diverges and appears to come from the focal point.
  • Ray striking the center of curvature (C): This ray reflects back along its original path in both types of mirrors.


I, Cronholm144, CC BY-SA 3.0, via Wikimedia Commons

9.2 - गोलाकार दर्पण (Spherical Mirrors)

गोलाकार दर्पण (Spherical Mirrors) वे दर्पण होते हैं जो गोलाकार (sphere) आकार के होते हैं। ये दो प्रकार के होते हैं:

  • धनात्मक दर्पण (Concave Mirror): यह दर्पण अंदर की ओर घुमावदार होता है, यानी यह एक अंदर मुड़ा हुआ क्षेत्र होता है। इसे अवतल दर्पण भी कहा जाता है।
  • नकारात्मक दर्पण (Convex Mirror): यह दर्पण बाहर की ओर घुमावदार होता है, यानी यह एक बाहर मुड़ा हुआ क्षेत्र होता है। इसे उत्तल दर्पण भी कहा जाता है।

गोलाकार दर्पण का मुख्य घटक उसकी सतह (surface) है, जो गोलाकार रूप में होती है। इन दर्पणों की विशेषताएँ और उपयोग विभिन्न स्थानों पर किए जाते हैं, जैसे कि दूरबीनों में, वाहन साइड मोल्ड्स में, और अपातकालीन स्थितियों में।

गोलाकार दर्पणों के महत्वपूर्ण घटक (Important Elements of Spherical Mirrors):
  • केंद्रीय बिंदु (Centre of Curvature, C): यह बिंदु गोलाकार दर्पण के गोलाकार सिरे पर स्थित होता है।
  • ध्रुव (Pole, P): यह दर्पण की सतह का वह बिंदु होता है, जिसे दर्पण की मुख्य सतह के केंद्र के रूप में माना जाता है।
  • आधिकारिक धुरी (Principal Axis): यह वह रेखा है जो केंद्र से दर्पण की सतह के केंद्र (P) तक जाती है और दर्पण के गोलाकार आकार के मध्य से गुजरती है।
  • फोकस (Focus, F): यह बिंदु वह स्थान होता है जहां पर सभी परावर्तित किरणें एकत्र होती हैं।
  • अपवर्तनित किरण (Reflected Ray): यह वह किरण है जो दर्पण की सतह से टकराने के बाद परावर्तित होती है।
गोलाकार दर्पणों का समीकरण (Mirror Formula):

गोलाकार दर्पणों के लिए एक सामान्य समीकरण होता है जो निम्नलिखित है:

$$ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $$

यहां,

  • f: फोकस की दूरी (Focus Distance)
  • u: वस्तु की दूरी (Object Distance)
  • v: प्रतिबिंब की दूरी (Image Distance)

गोलाकार दर्पणों की परावर्तित किरणें (Reflected Rays of Spherical Mirrors):

गोलाकार दर्पणों के बारे में एक और महत्वपूर्ण बिंदु यह है कि परावर्तित किरणें एक विशेष नियम का पालन करती हैं। यह नियम निम्नलिखित हैं:

  • पहला नियम: यदि कोई रेखा दर्पण की धुरी के समांतर आती है, तो यह रेखा फोकस में परावर्तित होती है (धनात्मक दर्पण के लिए) या फोकस से परावर्तित होती है (नकारात्मक दर्पण के लिए)।
  • दूसरा नियम: यदि कोई रेखा दर्पण के केंद्र से आती है, तो यह रेखा दर्पण के समानांतर परावर्तित होती है।
गोलाकार दर्पणों के प्रकार (Types of Spherical Mirrors):

गोलाकार दर्पणों के दो मुख्य प्रकार होते हैं:

  • धनात्मक दर्पण (Concave Mirror): यह दर्पण अंदर की ओर मुड़ा होता है और इसके द्वारा उत्पन्न प्रतिबिंब वास्तविक या काल्पनिक हो सकते हैं।
  • नकारात्मक दर्पण (Convex Mirror): यह दर्पण बाहर की ओर मुड़ा होता है और इसके द्वारा उत्पन्न प्रतिबिंब हमेशा काल्पनिक और सीधे होते हैं।

इन दर्पणों के उपयोग विभिन्न स्थानों पर होते हैं, जैसे कि वाहन साइड मोल्ड्स, दवाइयों के परीक्षण उपकरणों में आदि।

Define the principal focus of a concave mirror.

Solution:

Principal Focus of a Concave Mirror

The principal focus of a concave mirror is the point where parallel rays of light, striking the mirror surface, converge after reflection. This point lies on the principal axis, which is the line perpendicular to the mirror's surface at its center. The distance from the mirror's surface to the principal focus is called the focal length. In concave mirrors, the principal focus is located in front of the mirror, making it a converging mirror. This property is utilized in applications such as reflecting telescopes and makeup mirrors to produce magnified images.

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The radius of curvature of a spherical mirror is 20 cm. What is its focal 
length?

Solution:

Focal Length of a Spherical Mirror

The radius of curvature (R) of a spherical mirror is related to its focal length (f) by the formula:

f = R/2

Given the radius of curvature is 20 cm, we can substitute this value into the formula:

f = 20 cm / 2 = 10 cm

Thus, the focal length of the spherical mirror is 10 cm.

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Name a mirror that can give an erect and enlarged image of an object.

Solution:

Mirror That Gives Erect and Enlarged Image

A concave mirror is the type of mirror that can produce an erect and enlarged image of an object. When the object is placed between the focal point and the mirror, the light rays converge, forming a virtual image. This image appears upright and larger than the actual object. Concave mirrors are commonly used in applications such as makeup mirrors and shaving mirrors, where a larger, upright image is desirable for better visibility. The unique curvature of the concave surface allows for this magnification effect.

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Why do we prefer a convex mirror as a rear-view mirror in vehicles?

Solution:

Reasons for Using Convex Mirrors as Rear-View Mirrors

Convex mirrors are preferred as rear-view mirrors in vehicles due to their ability to provide a wider field of view. This is because they curve outward, allowing drivers to see more of the area behind them, including blind spots. The image formed is virtual, diminished, and upright, making it easier to judge distances. Additionally, convex mirrors reduce the risk of accidents by enabling drivers to monitor surrounding traffic more effectively. Their compact size also makes them easy to install in vehicles without obstructing visibility.

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Find the focal length of a convex mirror whose radius of curvature is 32 cm.

Solution:

Focal Length of a Convex Mirror

The focal length (f) of a convex mirror can be calculated using the formula:

f = R/2

where R is the radius of curvature. Given that the radius of curvature (R) is 32 cm, we can substitute this value into the formula:

f = 32 cm / 2 = 16 cm

Since the focal length of a convex mirror is considered positive, the focal length is +16 cm.

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A concave mirror produces three times magnified (enlarged) real image of an object placed at 10 cm in front of it. Where is the image located?

Solution:

Location of the Image in a Concave Mirror

For a concave mirror, the magnification (m) is given by the formula m = -v/u, where v is the image distance and u is the object distance. Given that the object distance (u) is -10 cm (negative as per sign convention) and the magnification (m) is 3, we can rearrange the formula:

3 = -v / (-10) => v = 30 cm.

The image is located 30 cm in front of the mirror, which means it is a real and inverted image.

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