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**Lenses**

A lens is a piece of any transparent material bounded by two curved surfaces or by one curved and one plane surface. Lens are of two types :

**(i) Convex or convergent lens:**

**(ii) Concave or divergent lens. **

**Optical Centre :** O is a point for a given lens through which any ray passes undeviated.

**Principal Axis : C _{1 }C_{2 }** is a line passing through optical centre and perpendicular to the lens.

**Principal Focus :** A lens has two surfaces and hence two focal points. First focal point is an object point on the principal axis for which image is formed at infinity.

While second focal point is an image point on the principal axis for which object lies at infinity.

**Focal Length f **is defined as the distance between optical centre of a lens and the point where the parallel beam of light converges or appears to converge.

**Aperture :** In reference to a lens, aperture means the effective diameter. Intensity of image formed by a lens which depends on the light passing through the lens will depend on the square of aperture, i.e., I Âµ (Aperture)^{2 }

**Rules for Image Formation:**

- A ray passing through optical centre proceeds undeviated through the lens.
- A ray passing through first focus or directed towards it, after refraction from the lens, becomes parallel to the principal axis.
- A ray passing parallel to the principal axis after refraction through the lenses passes or appears to pass through F
_{2}._{ }

**For Convergent or Convex Lens:**

Object | Image | Magnification |

âˆž | F | m << â€“ 1 |

âˆž â€“ 2F | F â€“ 2F | m < â€“1 |

2F | 2F | m = â€“1 |

F â€“ 2F | âˆž â€“ 2F | m > â€“1 |

F | âˆž | m >> â€“1 |

F â€“ O | In front of lens | m > + 1 |

**Image formation for convex lens (Convergent lens):**

(i) Object is placed at infinity.

Image : at F

real ,inverted

very small in size

m << - 1

(ii) Object is placed in between âˆž -2F.

Image : real (F - 2F)

inverted

small in size

(diminished)

m < - 1

(iii) Object is placed at 2F.

Image : real (at 2F)

inverted

equal (of same size)

(m = -1)

(iv) Object is placed in between 2F - F.

Image : real (2F - âˆž)

inverted

enlarged

m > 1

(v) Object is placed at F.

(vi) Object is placed in between F - O.

Image : virtual (in front of lens)

erected

enlarge

(m > + 1)

Given | Real | Virtual |

u | - | - |

v | + | - |

h_{1} | + | + |

h_{2} | - | + |

m | - | + |

**Image formation for concave lens (divergent lens):**

Imge is virtual, diminished, erect, towards the object, m = **+ve**

(i) Object is placed at infinity.

Image : At F

virtual

erected

diminished

(m << + 1)

(ii) Object is placed infront of lens.

Image : between F and optical centre

virtual

erected

diminished

(m < + 1)

**Formula of lens :** Focal length of a lens can be find out by the following formula:

Where - f = Focal length of lens.

v = Distance of image from pole

u = Distance of object from pole.

**Uses of this formula:**

(i) Put the correct signs of known variables according to the sign conventions.

(ii) Do not put the sign of unknown variable. The sign will automatically show up during calculations.

(iii) If the calculated sign of a variable turns out positive, then the variable calculated is on the other side of the lens, i.e., on the opposite side to the object. However if calculated variable is of negative sign, then it is on the same side as the object.

**Combination of lenses:**

**Two thin lens are placed in contact to each other:**

power of combination.

P = P_{1} + P_{2}

Use sign convention when solving numericals.

**Two thin lens are placed in at a small distance d.**

_{ }

Use sign convention when solving numericals.

**Power of lens** **:** The power of a lens is defined as reciprocal of focal length of the lens. Focal length should always be measured in meters.

Unit of power of lens is 1/meter which is called **dioptre.**

**Magnification of lens : **The magnification is defined as the ratio of the height of the image and the height of the object it is represented by M.

**Uses of lenses in daily life:**

Lenses are using is microscope, telescope. prism, binoculars and slide projector etc. **Relation between focal length (f) and Radius of curvature (R):**

**Focal length of a lens depends on following factors :-**

(i) Refractive index of the material of the lens.

(ii) Radius of curvature R_{1} and R_{2} of both in curved surfaces of the lens.

(iii) Colour (wave length) of the light.

Formula of making a spherical lens of specific focal length.

Where f is the average value of focal length for all the colours. Âµ is the refractive index of the material of the lens with respect to air. R_{1} and R_{2} are the radius of curvature of the curved surfaces respectively. R_{1} is positive and R_{2} is negative for convex lens. R_{1} is negative and R_{2} positive for concave lens.

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