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SMA Physics โ€ข Physics Lab Class XII โ€ข Exp 16

๐Ÿชž Focal Length of Concave Mirror (uโ€“v Method)

๐Ÿ“– Manual
๐Ÿ“ Optical Bench Ray Tracing & Parallax Viewer
๐Ÿ’ก
Experiment Guide: Adjust the Object Needle Distance ($u$). Move the Image Needle ($v_{\text{needle}}$) until it coincides with the tip of the inverted real image. Slide the Observer Eye Position left-right to verify that parallax is zero!
Object Distance $u$: 30.0 cm
Theoretical Image $v$: 30.0 cm
Image Needle Pos: 30.0 cm
Parallax Status: ZERO PARALLAX โœ“
๐ŸŽ›๏ธ Optical Bench Upright Controls Focal Length $f = 15.0\text{ cm}$
Object Needle Distance ($u$) 30.0 cm
Range: 18.0 cm to 60.0 cm (Beyond $f$) $u > f$ for Real Image
Image Needle Upright Position ($v_{\text{needle}}$) 30.0 cm
Adjust to coincide with real image tip Remove Parallax
Observer Eye Transverse Shift (Parallax Test) Center
โ† Left Tilt Direct Center Right Tilt โ†’
๐Ÿ“ Computed Optical Parameters
Focal Length $f = \frac{uv}{u+v}$
15.0 cm
Radius of Curvature $R = 2f$
30.0 cm
Linear Magnification $m = -v/u$
-1.00
Image Nature
Real & Inverted

Observation Table: Focal Length of Concave Mirror

Rough focal length $f_0 \approx 15\text{ cm}$. Formula: $f = \frac{uv}{u+v}$ (taking absolute magnitudes).

S.No. Object Distance $u$ (cm) Image Distance $v$ (cm) $1/u$ (cmโปยน) $1/v$ (cmโปยน) Focal Length $f = \frac{uv}{u+v}$ (cm) Parallax Check
๐Ÿ“Š Experimental Mean Result
Mean Focal Length ($f$)
-- cm
Mean Radius of Curvature ($R = 2f$)
-- cm

Optics Graphs: $u - v$ Curve & $1/u - 1/v$ Straight Line

Linear Equation: $\frac{1}{v} = -\frac{1}{u} + \frac{1}{f}$. Intercepts on both axes equal $\frac{1}{f}$.

โ„น๏ธ
The $u-v$ graph forms a rectangular hyperbola symmetric about the line $u = v = 2f$. The $1/u$ vs $1/v$ graph is a straight line having slope $-1$ and intercepts equal to $1/f = 0.067\text{ cm}^{-1}$ ($f = 15.0\text{ cm}$).

๐ŸŽฏ Aim of the Experiment

To find the value of $v$ for different values of $u$ in case of a concave mirror and to find its focal length ($f$).

๐Ÿ“ Governing Theory & Sign Convention

According to the Mirror Formula relating object distance $u$, image distance $v$, and focal length $f$:

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

By Cartesian sign conventions, for a concave mirror forming a real inverted image:

  • Light travels from left to right.
  • Object distance $u$ is negative (measured against incident light).
  • Real image distance $v$ is negative.
  • Focal length $f$ is negative.
$$\frac{1}{-f} = \frac{1}{-v} + \frac{1}{-u} \implies \frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

Parallax Removal Technique: When viewing the object needle and its real inverted image together from a slight lateral distance, if both needle tips shift relative to each other, parallax is present. By moving the image needle along the bench until both tips move together without lateral displacement, the needle exactly marks the real image position $v$.

โ“ Interactive Viva Voce Preparation