What is the Newton's formula for spherical mirror

Question Id: PH0000004

Question: What is the Newton's formula for spherical mirror?

Solution: 

A useful formula for spherical mirror is Newton's formula

Here "O" is the object and "I" is the image of the object. Let the distance of the object from the pole is \(u\) and distance of the object from the pole is \(v\) . 

Let the distance of the object and the image from the principal focus be \(x\) and \(y\) respectively.

So, \(x=u-f\Rightarrow u=f+x\)

And \(y=v-f\Rightarrow v=f+y\)

Using the value of \(u\) and \(v\) in the formula of mirror gives

\[\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\]

\[\Rightarrow \frac{1}{f+x}+\frac{1}{f+y}=\frac{1}{f}\]

\[\Rightarrow \frac{f+y+f+x}{(f+x)(f+y)}=\frac{1}{f}\]

\[\Rightarrow \frac{2f+y+x}{(f+x)(f+y)}=\frac{1}{f}\]

\[\Rightarrow 2f^{2}+fy+fx=f^{2}+fx+fy+xy\]

\[\Rightarrow 2f^{2}=f^{2}+xy\]

\[\Rightarrow f^{2}=xy\]

The above equation is the Newton's formula for Spherical mirror. As \(f\) is constant [ i.e \(f=\frac{r}{2}\) ] for a spherical mirror the the graph between  \(x\) and \(y\) is rectangular hyperbola.

Significance of Newtown's formula: the value of  \(f\) may be positive or negative but the  value of  \(f^{2}\) is always positive. That means \(xy\) is always positive which means the sign of \(x\) and \(y\) is always same. Which implies that object and his image will always lie on the same side of the focus.

 

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