Experiment: Finding Focal Length of Convex Lens and Concave Mirror - Tenth Grade Science
Determining the focal length of converging optical elements—convex lenses and concave mirrors—is a fundamental laboratory technique in geometric optics. In this practical investigation, students use distant object approximation to determine rough focal lengths, followed by optical bench ray tracing using the displacement method to measure focal lengths quantitatively using the lens and mirror formulas.
Experimental Aim and Objectives
The primary objectives of this practical investigation are:
1. To determine the approximate focal length of a concave mirror and a convex lens by focusing distant object rays onto a measurement screen.
2. To measure the object distance ($u$) and image distance ($v$) for various real image positions on an optical bench.
3. To calculate the precise focal lengths of the concave mirror and convex lens using the spherical mirror formula and thin lens equation.
4. To verify image formation characteristics (real, inverted, magnified, or diminished) across various object locations.
Theoretical Background and Optical Formulations
Both concave mirrors and convex lenses are converging optical elements that focus parallel incident rays onto a real principal focus.
DISTANT OBJECT APPROXIMATION
Parallel Rays from Distant Tree (u ~ Infinity)
==============================================>
\ Convex Lens
-----------------------------------------------\---+---------------
\ / \
X * Sharp Inverted Image
/ \ / formed at Screen
-----------------------------------------------/---+-- (Distance = f)
/
Rough Focal Length Determination
When an object is situated at a vast distance (effectively infinity, $u \approx \infty$), incident rays striking the optical element are essentially parallel to the principal axis.
According to geometric optics, parallel rays converge to form a sharp, real, inverted, point-like image located precisely in the focal plane ($v = f$). Measuring the linear separation between the optical center (or mirror pole) and the screen provides an initial rough estimate of the element's focal length.
The Spherical Mirror Formula
For a concave mirror, the object distance ($u$), image distance ($v$), and focal length ($f$) are related by:
$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f} \implies f = \frac{u v}{u + v}$$
Applying the New Cartesian Sign Convention: real objects sit in front of the mirror ($u < 0$), and real inverted images form in front of the mirror ($v < 0$). Consequently, the focal length of a concave mirror is always negative ($f < 0$).
The Thin Lens Formula
For a thin convex lens, the algebraic relation is:
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \implies f = \frac{u v}{u - v}$$
Under the Cartesian Sign Convention: real objects sit to the left of the lens ($u < 0$), while real inverted images form on the opposite side to the right ($v > 0$). Consequently, the focal length of a convex lens is always positive ($f > 0$).
Laboratory Apparatus
OPTICAL BENCH ALIGNMENT SCHEMATIC
[ Illuminated Object ] [ Convex Lens ] [ Screen ]
Object Upright Lens Holder Screen Upright
( 0 cm ) ( 50 cm ) ( 80 cm )
| | |
+------*-----------------------*--------------------*------+
| . . . . . . . . . . . . . . |
+----------------------------------------------------------+
Graduated Optical Rail (0 to 100 cm)
Required Equipment
Optical Bench: A rigid, graduated metal track (1.5 to 2.0 meters long) with sliding upright carriers. Convex Lens: Double convex optical glass lens (focal length roughly $+15 \text{ cm}$ to $+20 \text{ cm}$). Concave Mirror: Polished spherical concave mirror (focal length roughly $-15 \text{ cm}$ to $-20 \text{ cm}$). Illuminated Object: An optical lamp-box housing an illuminated wire-gauze cross-wire pattern, or a mounted optical pin. Image Screen: A white frosted glass or card screen mounted on an adjustable vertical carrier. Adjustable Lens and Mirror Upright Holders. Half-meter wooden scale and spirit level.
Part A: Determining the Focal Length of a Concave Mirror
CONCAVE MIRROR BENCH ALIGNMENT
Screen with Central Aperture
or Offset Screen
|
v
+-----------+
| [Screen] | Concave Mirror
+-----+-----+ .-----.
| / \
| Light from | P | Pole
| Lamp Box | | |
v \ /
+---------------+ '-----'
| [Lamp Object] | ^
+---------------+ |
|<------------- v ---------->|
|<------------- u ---------->|
Step-by-Step Procedure
- Rough Focal Length Estimate: Stand near an open window and direct the concave mirror toward a distant tree or utility pole. Position a white card in front of the mirror, shifting it back and forth until a sharp, inverted image of the distant landmark appears on the card. Measure the distance from the mirror surface to the card with a metric ruler. Record this as the rough focal length ($f_{\text{rough}}$).
- Align the optical bench horizontally using a spirit level.
- Mount the concave mirror in an upright holder near the 100 cm end of the optical bench rail. Note its bench position ($x_M$).
- Mount the illuminated wire-mesh object near the opposite end of the bench ($x_O$), positioning it beyond twice the rough focal length ($u > 2f$).
- Mount the white screen carrier between the object and the mirror, offsetting it slightly so it does not block light from the lamp box reaching the mirror.
- Slide the screen carrier along the rail until a sharp, clear, inverted image of the illuminated wire gauze is projected onto the screen.
- Record the position of the mirror ($x_M$), object ($x_O$), and screen ($x_S$) on the graduated bench scale.
- Calculate the object distance $u = x_M - x_O$ and image distance $v = x_M - x_S$.
- Move the object to four different positions (such as beyond $2f$, at $2f$, between $f$ and $2f$), adjusting the screen to find a sharp focus at each setting and recording all values in the observation table.
Data Log for Concave Mirror
CONCAVE MIRROR OBSERVATION TABLE
Trial Object Pos (cm) Mirror Pos (cm) Screen Pos (cm) u (cm) v (cm) f (cm)
---------------------------------------------------------------------------------
1 30.0 80.0 56.0 -50.0 -24.0 -16.2
2 40.0 80.0 52.0 -40.0 -28.0 -16.5
3 48.0 80.0 48.0 -32.0 -32.0 -16.0
4 55.0 80.0 42.5 -25.0 -37.5 -15.0
Mean Focal Length Calculation:
$$f_{\text{mean}} = \frac{(-16.2) + (-16.5) + (-16.0) + (-15.0)}{4} \approx -15.9 \text{ cm}$$
The focal length of the concave mirror is determined to be approximately $-15.9 \text{ cm}$.
Part B: Determining the Focal Length of a Convex Lens
CONVEX LENS BENCH ALIGNMENT
Illuminated Object Convex Lens Screen
(Lamp Box) (Holder) (Carrier)
| | |
v v v
+---------+ +-------+ +---------+
| [===] | | ( ) | | [===] |
+----+----+ +---+---+ +----+----+
| | |
|<---------- u --------->|<-------- v -------->|
| | |
Step-by-Step Procedure
- Rough Focal Length Estimate: Direct the convex lens toward a distant tree through an open window. Move a white card screen behind the lens until a sharp inverted image appears. Measure the distance between the lens center and the screen to determine the rough focal length ($f_{\text{rough}}$).
- Mount the illuminated wire-mesh object upright near the 0 cm mark of the optical rail ($x_O = 0.0 \text{ cm}$).
- Mount the convex lens in an upright carrier near the 50 cm mark ($x_L = 50.0 \text{ cm}$).
- Mount the white screen on an upright carrier on the opposite side of the lens.
- With the lens positioned beyond twice the rough focal length ($u > 2f$), slide the screen carrier back and forth until the projected image of the wire gauze achieves maximum sharpness.
- Record the positions of the object ($x_O$), lens ($x_L$), and screen ($x_S$).
- Calculate the object distance $u = x_L - x_O$ and image distance $v = x_S - x_L$.
- Adjust the lens and screen across five different object distances ($u > 2f$, $u = 2f$, and $f < u < 2f$), recording readings at each setting.
- Calculate the focal length for each trial using the thin lens formula and determine the average value.
Data Log for Convex Lens
CONVEX LENS OBSERVATION TABLE
Trial Object Pos (cm) Lens Pos (cm) Screen Pos (cm) u (cm) v (cm) f (cm)
-------------------------------------------------------------------------------
1 10.0 60.0 85.0 -50.0 +25.0 +16.7
2 20.0 60.0 86.7 -40.0 +26.7 +16.0
3 28.0 60.0 92.0 -32.0 +32.0 +16.0
4 35.0 60.0 100.0 -25.0 +40.0 +16.0
5 40.0 60.0 110.0 -20.0 +50.0 +14.3
Mean Focal Length Calculation:
$$f_{\text{mean}} = \frac{(+16.7) + (+16.0) + (+16.0) + (+16.0) + (+14.3)}{5} = +15.8 \text{ cm}$$
The focal length of the convex lens is determined to be approximately $+15.8 \text{ cm}$.
Laboratory Precautions and Sources of Error
- Parallax and Optical Axis Alignment: Ensure the centers of the illuminated object, lens or mirror, and screen are aligned along the same horizontal optical axis. If the optical centers are misaligned vertically or horizontally, the formed image will be distorted and displaced off the screen center.
- Sharp Image Identification: Move the screen past the point of maximum sharpness in both directions before setting the final position. The human eye accommodates to slight blur, so finding the exact focal position requires bracketing the sharpest focus.
- Sign Convention Consistency: Always apply the Cartesian Sign Convention to measured distances. For a convex lens forming real images, $u$ is negative while $v$ and $f$ are positive.
- Distant Object Selection: When determining rough focal length, select an object located at least 15 to 20 meters away. If the object is too close, incident rays will not be parallel, causing the measured rough focal length to overestimate the true value.
Chapter Exercises and Solutions
Exercise 1: During a convex lens focal length experiment on an optical bench, an object is positioned at the 15.0 cm mark, the lens carrier is clamped at the 55.0 cm mark, and a sharp inverted image is captured on a screen positioned at the 95.0 cm mark. Calculate the object distance ($u$), the image distance ($v$), and the focal length of the lens, and compute the magnification of the image.
Solution 1: Determine the optical bench distances relative to the lens carrier: object distance $u = x_{\text{object}} - x_{\text{lens}} = 15.0 \text{ cm} - 55.0 \text{ cm} = -40.0 \text{ cm}$. Image distance $v = x_{\text{screen}} - x_{\text{lens}} = 95.0 \text{ cm} - 55.0 \text{ cm} = +40.0 \text{ cm}$. Applying the thin lens equation $1/v - 1/u = 1/f$ yields $1/(+40.0) - 1/(-40.0) = 1/f$, which simplifies to $1/40.0 + 1/40.0 = 2/40.0 = 1/20.0 = 1/f$, giving $f = +20.0 \text{ cm}$. Because $u = -40.0 \text{ cm} = -2f$ and $v = +40.0 \text{ cm} = +2f$, the object is situated at twice the focal length. The linear magnification is $m = v / u = (+40.0 \text{ cm}) / (-40.0 \text{ cm}) = -1.0$. The negative sign confirms the image is real and inverted, while $|m| = 1.0$ indicates it is identical in size to the object.
Exercise 2: Explain why an experimenter cannot project a real, sharp image onto a viewing screen if an illuminated object is placed 10 cm in front of a convex lens having a focal length of 15 cm. Describe what the experimenter would see instead when looking directly through the lens.
Solution 2: A convex lens produces a real image only when the object distance is greater than its focal length ($u > f$). In this setup, the object is placed at $u = 10 \text{ cm}$, which is inside the focal length ($u < f = 15 \text{ cm}$). When rays diverge from a point inside the principal focus, the lens cannot converge them; the refracted rays emerge from the opposite side still diverging. Because diverging rays never physically intersect in space, no real image can be formed on a physical screen. However, if an observer looks through the lens from the opposite side, the eye projects the diverging rays backward to a virtual intersection behind the object on the same side. The observer sees an enlarged, upright, virtual image—the lens functions as a simple magnifying glass.
Exercise 3: When performing a focal length experiment with a concave mirror, a student notes that when the illuminated object pin and the viewing screen are positioned at the exact same distance from the mirror surface, the inverted image appears superimposed on the object with identical dimensions. Explain the geometric significance of this configuration, and deduce the mirror's focal length if this coincidence occurs at a distance of 36.0 cm.
Solution 3: In geometric spherical mirror optics, whenever an object is situated at the Center of Curvature ($C$), rays striking the mirror reflect back along their incident paths because they strike the spherical surface normal to the mirror tangent. Consequently, the reflected rays converge to form a real, inverted image at the same position ($C$), with a magnification of $m = -1.0$. The condition where the object and image coincide at the same distance confirms that both are located at the center of curvature ($u = v = R = 36.0 \text{ cm}$). Because the focal length ($f$) of a spherical mirror equals half its radius of curvature ($f = R / 2$), the focal length is $f = -36.0 \text{ cm} / 2 = -18.0 \text{ cm}$.