Helmholtz Relation

M’N’ is the real inverted image of the object MN. Consider the incident ray NA, which after refraction passes through N’.

Let the refracted An’ be inclined at an angle θ2 to the axis.

In the paraxial region, angular magnification



Linear magnification



Or, u₁y₁ tan θ₁ = u₂y₂ tan θ₂ = constant                  (i)

This is known as Langrange’s law.

If θ₁ and θ₂ are small then u₁y₁ sin θ₁ = u₂y₂ sin θ₂               (ii)

This is known as Abbe’s sine condition.

Langrange’s law can be used for a number of coaxial refracting surfaces and

u₁y₁θ₁ = u₂y₂θ₂ = u₃y₃θ₃ ……. = u_ny_nθ_n

where y_n is the size of the final image formed by the (n – 1)^th refracting surface and situated in a medium of refractive index u_n.

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