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Question

What is len's formula? Derive the formula 1v1u=1f for spherical lens.

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Solution

Lens formula: An equation which relates the object distance, image distance and focal length of a lens is called the lens formula. It is given as
1v1u=1f
The figure shows the formation of a real, inverted and diminished image AB of the object AB placed beyond the centre of curvature at a distance u from the convex lens.
Let v be the image distance.
According to Cartesian sign convention,
Object distance (OB)=u
Image distance (OB)=+v
Focal length (OF1=OF2)=+f
From the geometry of the figure above, right angled ABO and ABO are similar.
ABAB=OBOB=uv....(1)
Similarly, from the geometry of the figure, right angled ODF2 and BAF2 are similar.
ODAB=OF2F2B
ABAB=OF2F2B(OD=AB)
ABAB=OF2OBOF2
ABAB=fvf....(2)
From (1) and (2),
uv=fvf
u(vf)=vf
uv+uf=vf
Dividing each term by uvf,
uvuvf+ufuvf=vfuvf
1f+1v=1u
1v1v=1f
This equation is the 'Lens formula'.

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