Magnification

It is the process of making something look larger so that small details, which are usually too tiny to see with the naked eye, become visible. It’s commonly used in science—especially in fields like biology and materials science—where researchers need to closely examine things like cells, tissues, or the surfaces of materials. Tools like microscopes or magnifying glasses help enlarge these tiny structures, allowing scientists to see and study their finer details. By zooming in, magnification helps us explore and better understand the microscopic world that would otherwise stay hidden.

1.0Definition of Magnification

• It is the optical process of enlarging the apparent size of an object, enabling detailed examination of small or distant subjects. It is quantified by the ratio of the image size to the object size.

$m=\frac{\text{ Height of image }}{\text{ Height of Object }}=\frac{v}{u}$

where v is the image separation and u is the object distance.

2.0Types of Magnification

1.Linear Magnification: This describes how much bigger or smaller an image is compared to the actual object, based on their sizes measured straight across (not along the line of sight). If the value is negative, the image is upside down; if it's positive, the image is right side up.

2.Angular Magnification:This measures the angle subtended by the image at the observer's eye compared to the angle subtended by the object when viewed without magnification. It is commonly used in instruments like telescopes and microscopes.

3.0Types of Linear Magnification

1.Positive Linear Magnification: Occurs when the image is upright and virtual. This is typically seen in instruments like magnifying glasses or when using a concave lens.

2.Negative Linear Magnification: Indicates that the image is inverted. This is common in real images formed by convex lenses or concave mirrors.

3.Unitary Linear Magnification: When the image size equals the object size, resulting in a magnification factor of 1. This is often the case with plane mirrors.

4.0Types of Angular Magnification

Visual angle: Visual angle is the angle, a viewed object or image subtends at the eye. It is also called the object’s angular size.

Nearer the object larger the visual angle then the object appears big in size.

Note: Near point (N.P.) of normal eye D = 25cm

Far point (F.P.) of normal eye = ∞

Angular Magnification ‘OR’ Magnifying power

$\text{ M.P. }=\frac{\text{ visual angle of image in the presence of instrument }(\beta )}{\text{ maximum visual angle of an object in the absence of instrument }(\alpha )}$

1.Simple Magnifier (Magnifying Glass):

• A single converging (convex) lens with a short focal length, commonly known as a magnifying glass, forms a virtual, erect, and magnified image when the object is placed between its focal point and optical center.

$\tan \beta =\frac{h_0}{u}=\frac{h_i}{v}$

If angle is small

$\begin{array}{rl}& \beta =\frac{h_0}{u}=\frac{h_i}{v}\\ & \tan \alpha \approx \alpha =\frac{h_0}{D}\\ & M\cdot P=\frac{\beta }{a}=\frac{\frac{h_0}{u}}{\frac{h_0}{D}}\\ & M\cdot P=\frac{D}{|u|}\end{array}$

Case (1): When image formed at a distance of least distance of distinct vision from the lens.

$\begin{array}{rl}& \frac{1}{-D}-\frac{1}{-u}=\frac{1}{f}\Rightarrow \frac{1}{u}=\frac{1}{D}+\frac{1}{f}\\ & [v=-D\text{ and }u=-ve]\end{array}$

Multiplying both the sides by D

$\begin{array}{rl}& \frac{D}{u}=1+\frac{D}{f}\Rightarrow MP=\frac{D}{u}=1+\frac{D}{f}\\ & (\text{ M.P. }{)}_{\max }=\frac{D}{u_{\min }}=1+\frac{D}{f}\end{array}$

Case (2): When image formed at infinite distance from the lens.

From Lens Equation

$\begin{array}{rl}& \frac{1}{v}-\frac{1}{u}=\frac{1}{f}\Rightarrow \frac{1}{-\infty }-\frac{1}{-u}=\frac{1}{f}\Rightarrow u=f\\ & \text{ So M.P. }=\frac{D}{u}=\frac{D}{f}\\ & \text{ (M.P.) }{}_{\min }=\frac{D}{u_{\max }}=\frac{D}{f}\end{array}$

2.Compound microscope

• A compound microscope is a high-magnification laboratory instrument with multiple lenses, used to study detailed structures of cells, tissues, or organs, and can magnify objects up to 1000 times

Magnifying Power: 

Total magnifying power = Linear magnification of objective lens × angular magnification MP of eye lens

$\begin{array}{rl}& \text{ M.P. }=m_0\times m_e\\ & \text{ M.P. }=-\left|\frac{v_0}{u_0}\frac{D}{u_e}\right|\end{array}$

Case (1): When final image formed at least distance of distinct vision.

$\text{ M.P. }=\frac{v_0}{u_0}\left[1+\frac{D}{f_e}\right]=\frac{f_0}{\left(f_0+u_0\right)}\left[1+\frac{D}{f_e}\right]=\frac{f_0-v_0}{f_0}\left[1+\frac{D}{f_e}\right]$

Distance between both the lenses, $L=v_0+\left|u_e\right|$

Case (2): When the final image formed at infinity.

$\text{ M.P. }=\frac{v_0}{u_0}\left[\frac{D}{f_e}\right]=\frac{f_0}{\left(f_0+u_0\right)}\left[\frac{D}{f_e}\right]=\frac{f_0-v_0}{f_0}\left[\frac{D}{f_e}\right]$

Distance between both the lenses, $L=v_0+f_e$

Note:Sign convention for solving numericals, $u_0=-ve,v_0=+ve$

$f_0=+ve,u_e=-ve,v{}_e=-ve,f_e=+ve,m_0=-ve,m_e=+ve,M.P.=-ve$

Magnifying power of compound microscope when final image formed at infinity

$\begin{array}{rl}& \tan \theta =\frac{h_0}{f_0}\\ & \tan \theta =\frac{h_i}{l}\\ & \Rightarrow \frac{h_0}{f_0}=\frac{h_i}{l}\Rightarrow \frac{h_i}{h_0}=\frac{l}{f_0}\end{array}$

So, M.P. of compound microscope when final image formed at ∞

$\begin{array}{rl}& \text{ M.P. }=m_0\times m_e\\ & \text{ M.P. }=-\frac{l}{f_0}\times \frac{D}{f_e}\end{array}$

3.Astronomical telescope

• An astronomical telescope uses an objective with a long focal length and wide aperture, along with an eyepiece, to provide angular magnification of distant objects, forming a real image that the eyepiece magnifies into a final inverted image.

$\begin{array}{rl}& \alpha =\frac{h}{f_0}\\ & \beta =-\frac{h}{u_e}\end{array}$

Magnifying Power

$\begin{array}{rl}& \text{ M.P. }=\frac{\text{ visual angle with instrument }(\beta )}{\text{ visual angle for unaided eye }(\alpha )}\Rightarrow \text{ M.P. }=\frac{-\frac{h}{u_e}}{\frac{h}{f_0}}=-\frac{f_0}{f_e}\\ & \text{ M.P. }=-\frac{f_0}{f_e}\end{array}$

Case (1): When final image formed at least distance of distinct vision

$\begin{array}{rl}& \frac{1}{-D}-\frac{1}{-u_e}=\frac{1}{f_e}\Rightarrow \frac{1}{u_e}=\frac{1}{f_e}+\frac{1}{D}=\frac{1}{f_e}\left[1+\frac{f_e}{D}\right]\\ & \text{ M.P. }=-\frac{f_0}{u_e}=-\frac{f_0}{f_e}\left[1+\frac{f_e}{D}\right]\end{array}$

Length of the tube is, $L=f_0+\left|u_e\right|$

Case (2): When final image formed at infinity

$\begin{array}{rl}& \frac{1}{-\infty }-\frac{1}{-u_e}=\frac{1}{f_e}\Rightarrow u_e=f_e\\ & \text{ So M.P. }=-\frac{f_0}{f_e}\end{array}$

Length of the tube is, $L=f_0+f_e$

5.0Transverse or Lateral Magnification

• If a one dimensional object is placed perpendicular to the principal axis then ratio of image height and object height is called transverse or lateral magnification.

$m_t=\frac{h_i}{h_o}$

• If $m_t$ is positive means erect image is formed

•  If $m_t$ is negative means inverted image is formed

$\text{ 1. }\left|m_t\right|>1$

$\text{ 2. }\left|m_t\right|<1$

$\text{ 3. }\left|m_t\right|=1$

Derivation of Transverse Magnification:

$\frac{h_i}{h_o}=\frac{v}{u}$

Put the value with sign,

$\begin{array}{rl}& \frac{-h_i}{h_o}=\frac{-v}{-u}\\ & \frac{h_i}{h_o}=-\frac{v}{u}\\ & m_t=\frac{h_i}{h_o}=-\frac{v}{u}\\ & \tan \theta =\frac{h_o}{u}\\ & \tan \theta =\frac{h_i}{v}\\ & \frac{h_i}{h_o}=\frac{v}{u}\end{array}$

6.0Longitudinal Magnification

• If an object is placed along the principal axis then the ratio of length of image and length of object is called longitudinal or axial magnification.

$m_L=\frac{\text{ Length of image }}{\text{ Length of Object }}=\frac{I_i}{I_o}$

$m_L=\frac{\text{ Length of image }}{\text{ Length of Object }}=\left|\frac{v_1-v_2}{u_2-u_1}\right|$

Longitudinal magnification for small objects $(Lo\ll f)$

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Differentiating w.r.t. U

$-\frac{1}{v^2}\frac{dv}{du}-\frac{1}{u^2}=0\Rightarrow \frac{dv}{du}=-\frac{v^2}{u^2}$

If we use only magnitude then $\frac{dv}{du}=-\frac{v^2}{u^2}$

$m_L=\frac{I_i}{I_o}=\frac{v^2}{u^2}=m_t^2$

7.0Superficial Magnification

• If a two-dimensional object is placed with its plane perpendicular to the principal axis then its magnification is known as superficial magnification.

$\begin{array}{rl}{m}_s& =\frac{A_i}{A_o}\\ A_o& =w_o{h}_o\text{ and }{A}_i=w_i{h}_i\\ m_t& =\frac{h_i}{h_o}=\frac{w_i}{w_o}\\ \frac{A_i}{A_o}& =\frac{w_i{h}_i}{w_o{h}_o}\\ m_t& =\frac{h_i}{h_o}=\frac{w_i}{w_o}\\ \frac{A_i}{A_o}& =m_t^2=m_s\end{array}$

8.0Volume Magnification:

• For small cubical objects, all dimensions are magnified equally because they are all approximately the same distance from the mirror. As a result, the final image retains the shape of a cube. Additionally, since the area of the object is oriented perpendicular to the principal axis, the magnification applies uniformly across the object's surface.

Area of image $\left(A_i\right)=m^2{A}_o$

Length of image (for small object) $\Rightarrow l_i=m^2{l}_o$

Then, volume of image $=\left(A_i\right)\times l_i=\left(m^2{A}_o\right)\left(m^2{l}_o\right)$

$\Rightarrow m^4\left(\begin{array}{ll}{A}_{0}l& \\ l_0& \end{array}\right)$

Volume of image $=m^4{V}_o$

Volume magnification $=\frac{\text{ Volume of image }}{\text{ Volume of Object }}=m^4$