91Ó°ÊÓ

$\begin{array}{rcl}f& =& 10\text{cm}\\ P_n& =& 25\text{cm}\\ & & \end{array}$

According to the given condition, the image appears at P_n;fromthat, find the object distance and, using the formula of the angular magnification, calculate the magnification when the image appears atP_n. Similarly, calculate the magnification when the image appears at ∞.

The formulas are as follows:

$$\begin{gathered} \frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\left(\frac{1}{p}+\frac{1}{i}\right) \\ \mathit{m}\mathbf{=}\frac{\mathbf{}{\mathbf{θ}}^{\mathbf{\text{'}}}}{\mathbf{θ}} \end{gathered}$$

Where p is the pole, f is the focal length, and i is the image distance.

(a)

Without the magnifier, we can write the angle from figure 34-19,

$\mathrm{θ}=h/P_n$

Where h is the original height of the object.

With magnifier,

$i=-\left|i\right|=-P_n$

So, now, according to the formula,

$\begin{array}{rcl}\frac{1}{f}& =& \left(\frac{1}{P}+\frac{1}{i}\right)\\ \frac{1}{P}& =& \left(\frac{1}{f}+\frac{1}{P_n}\right)\\ P& =& \frac{f{P}_n}{f+P_n}\\ & & \end{array}$

So, now the angular magnification is,

$\begin{array}{rcl}{m}_{\mathrm{θ}}& =& \frac{{\mathrm{θ}}^{\text{'}}}{\mathrm{θ}}\\ m_{\mathrm{θ}}& =& \frac{h/P}{h/P_n}\\ m_{\mathrm{θ}}& =& \frac{P_n}{P}\\ & & \end{array}$

By substituting the value,

$\begin{array}{rcl}{m}_{\mathrm{θ}}& =& \frac{P_n\left(f+P_n\right)}{f{P}_n}\\ m_{\mathrm{θ}}& =& \frac{\left(f+P_n\right)}{f}\\ m_{\mathrm{θ}}& =& 1+\frac{P_n}{f}\\ m_{\mathrm{θ}}& =& 1+\frac{25}{10}\\ & & \end{array}$

$m_{\mathrm{θ}}=3.5$

Therefore, the angular magnification of the image that appears at P_n is 3.5.

(b)

Now, consider the image appears at infinity so that,

$\begin{array}{rcl}i& =& -\left|i\right|=∞\\ \frac{1}{f}& =& \left(\frac{1}{P}+\frac{1}{i}\right)\\ \frac{1}{P}& =& \frac{1}{f}\\ P& =& f\end{array}$

So, the magnification is,

$\begin{array}{rcl}{m}_{\mathrm{θ}}& =& \frac{h/P}{h/P_n}\\ m_{\mathrm{θ}}& =& \frac{P_n}{P}\\ m_{\mathrm{θ}}& =& \frac{P_n}{f}\\ m_{\mathrm{θ}}& =& \frac{25}{10}\end{array}$

Therefore, the angular magnification of the image that appears at infinity is 2.5.

Using the corresponding formula, find the angular magnification of the image by a simple magnifying lens for different object distances.