Power of a Lens and Combination of Thin Lenses in Contact
- The power of a lens is defined as the reciprocal of its focal length.
- It is denoted by the symbol ‘P’ and can also be expressed in diopters (D).
- The formula to calculate the power is: P = 1/f
- Positive power denotes a converging lens, while negative power denotes a diverging lens.
- Power is a scalar quantity and its unit is diopters.
Combination of Several Lenses
When two or more thin lenses are placed in contact, their powers can be added algebraically to determine the power of the combination. Example:
- If a converging lens has a power of +3D and a diverging lens has a power of -2D, the power of the combination will be +3D + (-2D) = +1D. Note:
- The sign convention for power remains the same while adding powers of lenses.
Formulas for Combining Lenses in Contact
- For two lenses in contact:
- P_combination = P1 + P2
- For three lenses in contact:
- P_combination = P1 + P2 + P3
- For n lenses in contact:
- P_combination = P1 + P2 + P3 + … + Pn Where P_combination is the power of the combination and P1, P2, P3, …, Pn are the individual powers of the lenses.
Example of Combining Lenses
Suppose we have four lenses in contact: Lens A, B, C, and D.
- Lens A has a power of +4D
- Lens B has a power of -3D
- Lens C has a power of +2D
- Lens D has a power of -5D To find the power of the combination: P_combination = P_A + P_B + P_C + P_D = +4D + (-3D) + +2D + (-5D) = -2D Therefore, the power of the combination of these lenses is -2D.
Equivalent Focal Length
- The power of a combination of lenses in contact can also be used to determine the equivalent focal length.
- The formula to calculate the equivalent focal length is: f_combination = 1/P_combination Example:
- If the power of a combination of lenses is +3D, then the equivalent focal length will be 1/3 = 0.33 meters.
Magnification in Lens Combinations
- The magnification of a lens combination is the product of the magnifications of individual lenses.
- The formula to calculate the magnification is: m_combination = m1 x m2 x m3 x … x mn Example:
- If the magnifications of two lenses are 2 and 3 respectively, then the magnification of the combination will be 2 x 3 = 6.
Example of Lens Combination: Converging Lens and Diverging Lens
Consider a converging lens with a focal length of +10 cm and a diverging lens with a focal length of -5 cm. To find the overall power of the combination: P_combination = P_converging lens + P_diverging lens = 1/f_converging lens + 1/f_diverging lens = 1/10 + 1/(-5) = 1/10 - 1/5 = 1/10 - 2/10 = -1/10 D Therefore, the overall power of the combination is -1/10 D.
Example of Lens Combination: Two Converging Lenses
Suppose two converging lenses, Lens A and Lens B, are in contact.
- Lens A has a power of +4D and a focal length of +25 cm.
- Lens B has a power of +5D and a focal length of +20 cm. To find the overall power of the combination: P_combination = P_A + P_B = 4 + 5 = +9D Therefore, the overall power of the combination is +9D.
Example of Lens Combination: Two Diverging Lenses
Suppose two diverging lenses, Lens X and Lens Y, are in contact.
- Lens X has a power of -2.5D and a focal length of -40 cm.
- Lens Y has a power of -4.5D and a focal length of -22.2 cm. To find the overall power of the combination: P_combination = P_X + P_Y = -2.5 - 4.5 = -7D Therefore, the overall power of the combination is -7D.
Summary
- Power of a lens is the reciprocal of its focal length and is denoted by ‘P’ or in diopters (D).
- When combining lenses in contact, the powers can be added algebraically.
- The formula for combining two lenses is: P_combination = P1 + P2.
- Equivalent focal length can be calculated using the formula: f_combination = 1/P_combination.
- Magnification in lens combinations is calculated as the product of individual magnifications.
Power of a Lens and Combination of Thin Lenses in Contact - Example
Suppose we have two converging lenses in contact:
- Lens A has a power of +2D and a focal length of +50 cm.
- Lens B has a power of +3D and a focal length of +33.33 cm. To find the overall power of the combination:
- P_combination = P_A + P_B = 2 + 3 = +5D Therefore, the overall power of the combination is +5D.
Power of a Lens and Combination of Thin Lenses in Contact - Magnification
- Magnification can be calculated using the formula: m = -v/u (where m is the magnification, v is the image distance, and u is the object distance). Example:
- If the object distance is 20 cm and the image distance is 10 cm, then the magnification will be: m = -10/20 = -0.5.
Power of a Lens and Combination of Thin Lenses in Contact - Magnification in Lens Combinations
- The magnification in lens combinations is the product of individual magnifications. Example:
- Suppose a lens combination has two lenses with magnifications 2 and -3 respectively.
- The overall magnification will be: 2 x (-3) = -6.
Power of a Lens and Combination of Thin Lenses in Contact - Example
Consider a diverging lens with a power of -4D and a converging lens with a power of +6D. To find the overall power of the combination:
- P_combination = P_diverging lens + P_converging lens = -4 + 6 = +2D Therefore, the overall power of the combination is +2D.
Power of a Lens and Combination of Thin Lenses in Contact - Equivalent Focal Length
- The equivalent focal length in lens combinations can be calculated as: 1/f_combination. Example:
- If the equivalent focal length of a lens combination is 30 cm, then the power will be: P_combination = 1/f_combination = 1/30 = +0.0333 D.
Power of a Lens and Combination of Thin Lenses in Contact - Case of Diverging Lenses
- When combining diverging lenses in contact, their powers are added algebraically. Example:
- If two diverging lenses have powers of -3D and -2D respectively, then the overall power of the combination will be: P_combination = -3 + (-2) = -5D.
Power of a Lens and Combination of Thin Lenses in Contact - Case of Converging Lenses
- When combining converging lenses in contact, their powers are also added algebraically. Example:
- If two converging lenses have powers of +4D and +2D respectively, then the overall power of the combination will be: P_combination = 4 + 2 = +6D.
Power of a Lens and Combination of Thin Lenses in Contact - Multiple Lenses Example
Suppose we have three lenses in contact: Lens X, Lens Y, and Lens Z.
- Lens X has a power of +3D.
- Lens Y has a power of -2D.
- Lens Z has a power of +5D. To find the overall power of the combination:
- P_combination = P_X + P_Y + P_Z = 3 + (-2) + 5 = +6D Therefore, the overall power of the combination is +6D.
Power of a Lens and Combination of Thin Lenses in Contact - Magnification Formula
- The formula to calculate the magnification (m) is: m = height of image/height of object. Example:
- If the height of the image is 8 cm and the height of the object is 4 cm, then the magnification will be: m = 8/4 = 2.
Power of a Lens and Combination of Thin Lenses in Contact - Magnification in Lens Combinations Example
Suppose a lens combination has two lenses with magnifications 2 and -2 respectively. To find the overall magnification:
- Magnification = m1 x m2 = 2 x (-2) = -4 Therefore, the overall magnification of the combination is -4.
Combination of several lenses
- When multiple lenses are combined, the resulting power of the combination can be obtained by adding the individual powers of the lenses.
- The formula to calculate the power of the combination of several lenses is:
- P_combination = P1 + P2 + P3 + …
- Example:
- If a lens combination consists of three lenses, with powers +2D, -3D, and +4D, the overall power of the combination will be:
- P_combination = +2D + (-3D) + 4D = +3D
- If a lens combination consists of three lenses, with powers +2D, -3D, and +4D, the overall power of the combination will be:
Example of combining lenses
Suppose we have three lenses in contact: Lens A, Lens B, and Lens C.
- Lens A has a power of +2D.
- Lens B has a power of -3D.
- Lens C has a power of +5D. To find the overall power of the combination:
- P_combination = P_A + P_B + P_C = +2D + (-3D) + +5D = +4D Therefore, the overall power of the combination is +4D.
Equivalent focal length in lens combinations
- The equivalent focal length of a combination of lenses in contact can be calculated using the formula:
- 1/f_combination = 1/f1 + 1/f2 + 1/f3 + …
- Example:
- If a lens combination consists of two lenses with focal lengths +20 cm and -30 cm, the equivalent focal length can be calculated as:
- 1/f_combination = 1/f1 + 1/f2 = 1/20 + 1/(-30) = 0.05 - 0.0333 = 0.0167 cm
- If a lens combination consists of two lenses with focal lengths +20 cm and -30 cm, the equivalent focal length can be calculated as:
Example of equivalent focal length in lens combinations
Suppose a lens combination consists of three lenses: Lens X, Lens Y, and Lens Z.
- Lens X has a focal length of +10 cm.
- Lens Y has a focal length of -20 cm.
- Lens Z has a focal length of +30 cm. To find the equivalent focal length of the combination:
- 1/f_combination = 1/f_X + 1/f_Y + 1/f_Z = 1/10 + 1/(-20) + 1/30 = 0.1 - 0.05 + 0.0333 = 0.0833 cm Therefore, the equivalent focal length of the combination is 0.0833 cm.
Magnification in lens combinations
- Magnification is the ratio of the height of the image to the height of the object.
- In lens combinations, the magnification can be calculated by multiplying the individual magnifications of the lenses.
- The formula for calculating the magnification in lens combinations is:
- m_combination = m1 x m2 x m3 x …
- Example:
- If a lens combination has two lenses with magnifications 2 and -3 respectively, the overall magnification will be:
- m_combination = 2 x (-3) = -6
- If a lens combination has two lenses with magnifications 2 and -3 respectively, the overall magnification will be:
Example of magnification in lens combinations
Consider a lens combination with three lenses: Lens A, Lens B, and Lens C.
- Lens A has a magnification of 2.
- Lens B has a magnification of -3.
- Lens C has a magnification of 4. To find the overall magnification of the combination:
- m_combination = m_A x m_B x m_C = 2 x (-3) x 4 = -24 Therefore, the overall magnification of the combination is -24.
Example of combining lenses: Two converging lenses
Suppose we have two converging lenses in contact: Lens P and Lens Q.
- Lens P has a power of +3D.
- Lens Q has a power of +2D. To find the overall power of the combination:
- P_combination = P_P + P_Q = +3D + 2D = +5D Therefore, the overall power of the combination is +5D.
Example of combining lenses: Two diverging lenses
Suppose we have two diverging lenses in contact: Lens X and Lens Y.
- Lens X has a power of -4D.
- Lens Y has a power of -2D. To find the overall power of the combination:
- P_combination = P_X + P_Y = -4D + (-2D) = -6D Therefore, the overall power of the combination is -6D.
Example of combining lenses: Converging and diverging lenses
Suppose a lens combination consists of a converging lens and a diverging lens.
- The converging lens has a power of +3D.
- The diverging lens has a power of -2D. To find the overall power of the combination:
- P_combination = P_converging lens + P_diverging lens = +3D + (-2D) = +1D Therefore, the overall power of the combination is +1D.
Summary
- The power of a combination of lenses in contact can be obtained by adding the individual powers of the lenses.
- The equivalent focal length of a combination of lenses can be calculated by summing the reciprocals of the individual focal lengths.
- Magnification in lens combinations is calculated by multiplying the individual magnifications of the lenses.
- Examples have been provided to illustrate the concepts discussed.
Power of a Lens and Combination of Thin Lenses in Contact The power of a lens is defined as the reciprocal of its focal length. It is denoted by the symbol ‘P’ and can also be expressed in diopters (D). The formula to calculate the power is: P = 1/f Positive power denotes a converging lens, while negative power denotes a diverging lens. Power is a scalar quantity and its unit is diopters.