GEOMETRICAL OPTICS: Focal Length and the Index of Refraction

Q: A thin spherical planar-convex lens is surrounded by air:

The lens has a radius of curvature ( R₂ ) of 50mm and a refractive index ( n₁ ) of 1.50. Determine the focal length. Additionally, what will happen to the lens’ focal length if it is placed within a watery medium instead of air ( n_w = 1.33 )?

A: The information that is given will enable us to use the variation of the Thin-Lens Equation that relates focal length ( f ) to the physical attributes contained by the lens:

( 1 / f ) = ( n₁ – 1 )[ ( 1 / R₁ ) – ( 1 / R₂ ) ]

The problem is significantly simplified by the fact that a ray of light will first encounter a flat edge. The radius of curvature for such an edge is infinity:

R₁ = ∞

( 1 / R₁ ) = ( 1 / ∞ ) = 0

The lattermost edge will have a negative ( R₂ ) value. Furthermore, the equation is perfectly suited to solve problems dealing with a medium that consists of air. As we will see, however, the index of refraction ( n₁ ) will be altered if the lens is submerged in water. For now, let’s determine the focal length of the lens as it is surrounded by air:

( 1 / f ) = ( 1.50 – 1 )[ ( 1 / ∞ ) – ( 1 / – 0.050 m ) ]

( 1 / f ) = ( 0.50 )( 1 / 0.050 m )

( 1 / f ) = ( 0.50 )( 1 / 0.050 m )

( 1 / f ) = ( 0.50 )( 20 / m )

( 1 / f ) = ( 10 / m )

f = ( 1 m / 10 )

f = + 0.10 m

If the same lens is submerged into water, the index of refraction term in the Thin-Lens Equation must be substituted with a ratio of the prior refractive index value ( n₁ ) and that of water ( n_w ):

( 1 / f ) = [ ( n₁ / n_w ) – 1 ][ ( 1 / R₁ ) – ( 1 / R₂ ) ]

As we can see, the focal length of the lens will be increased as a consequence of being submerged within the newer medium. This is due to the lens having less of an ability to bring light rays into convergence than it did when surrounded by air:

( 1 / f ) = [ ( n₁ / n_w ) – 1 ][ ( 1 / R₁ ) – ( 1 / R₂ ) ]

( 1 / f ) = [ ( 1.50 / 1.33 ) – 1 ][ ( 1 / ∞ ) – ( 1 / – 0.050 m ) ]

( 1 / f ) = ( 1.13 – 1 )( 1 / 0.050 m )

( 1 / f ) = ( 0.13 )( 20 / m )

( 1 / f ) = ( 2.6 / m )

f = ( 1 m / 2.6 )

f = + 0.38 m