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Lights rays on the both upper and lower corner of the concave or convex lens show deflection. It is true . But I am trying to understand why light rays goes straight without bending while passing through the optical center of the lens. Is there anyone to make my problem solved.

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The light beam is refracted when it passes through the centre of the lens C, but it is not deflected into another direction. It emerges in the same direction with a small sideways displacement.

Any ray through C passes symmetrically through the lens, because the curvature of the faces is the same above and below the axis. The two faces of the lens are parallel where the ray enters and leaves, so this is like a ray passing through a rectangular block. The ray emerges parallel to its original direction but with a sideways displacement D.

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Ray diagrams and formulas for lenses are based on the paraxial approximation in which rays are always close to and parallel with the optical axis. Then the amount of displacement D is negligible, because it depends on the angle of incidence, which is assumed to be small. D also depends on thickness of the lens, which is likewise assumed to be small (the thin lens approximation). Because D is so small, the ray is drawn as a straight line through C, without a kink.

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The assertion is not quite true. Whilst the ray direction does not change for a ray through the optical center, there is a sideways translation.

The bigger the distance between the principal planes of the lens, the bigger the translation. This is one of the ways wherein the thin lens approximation is important. A thin lens is one whose principal plane can be taken to be co-incident. In that case, the lens's action can be thought of as happenning wholly on the unique, principal plane.

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That's not always true: it is only an approximation for thin lenses, but it is a really good one (it's been working everyday and it keeps working).

It is based on the paraxial approximation, for which, for small angles, you can use $\sin \theta \approx \tan \theta \approx \theta$. For more precision, you can take more terms in the Taylor development of the sine. That's called "3rd order optics (or higher)".

With this, you get the law of refraction as $n_1\cdot\epsilon_1\simeq n_2\cdot \epsilon_2$. Using this, you can deduce the Dioptre equation using the Abbe's invariant. The equation of a lens is obtained as the image through two dioptres together; and it shows our classical results.

For thicker lenses, you find that the equations are still valid, but measuring distances from the so called "principal planes". They can be seen as a "shifting" between the entrance and the exit of rays. That's why thin lenses are so useful (they're usually nice approximations).

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