In an experiment to measure the focal length (f) of a convex lens, the magnitude of object distance (x) and the image distance (y) are measured with reference to the focal point of the lens. The y-x plot is shown in figure.
The focal length of the lens is_____cm.
In an experiment to measure the focal length (f) of a convex lens, the magnitude of object distance (x) and the image distance (y) are measured with reference to the focal point of the lens. The y-x plot is shown in figure.
The focal length of the lens is_____cm.
Correct Answer: 20
Approach Solution - 1
From the lens formula:
\[\frac{1}{f} = \frac{1}{x} + \frac{1}{y}.\]
The plot shows the relationship between y (image distance) and x (object distance). For any specific point on the graph, the values of x and y can be substituted into the lens formula to calculate f.
Alternatively, the product of x and y at corresponding points can be used to directly calculate the focal length. From the lens equation, we know:
\[x \cdot y = f^2.\]
From the graph: - At point C, x = 20 cm and y = 20 cm.
Using \(x \cdot y = f^2\):
\[f^2 = 20 \cdot 20 = 400.\]
Taking the square root:
\[f = \sqrt{400} = 20 \text{ cm}.\]
Final Answer: The focal length of the lens is:
\[\boxed{20 \text{ cm}}.\]
Approach Solution -2
The problem provides a plot of the magnitude of image distance (y) versus the magnitude of object distance (x) for a convex lens. Both distances are measured with reference to the focal points of the lens. We need to determine the focal length (f) of the lens from this plot.
Concept Used:
When the object distance (x) and image distance (y) for a lens are measured from the respective focal points, their relationship with the focal length (f) is given by Newton's Lens Formula. The formula is:
\[ xy = f^2 \]This equation describes a hyperbolic relationship between x and y. To find the focal length, we can select a coordinate pair (x, y) from the experimental curve shown in the plot and substitute it into this formula.
Step-by-Step Solution:
Step 1: Identify the governing equation. According to the problem description, the distances x and y are measured from the focal points. Therefore, we use Newton's Lens Formula:
\[ xy = f^2 \]Step 2: Extract a data point from the given y-x plot. The curve represents the experimental data. Several points (A, B, C, D, E) are shown on the curve. Point C is clearly marked with dashed lines indicating its coordinates on the x and y axes.
From the graph at point C:
\[ x = 20 \, \text{cm} \] \[ y = 20 \, \text{cm} \]Step 3: Substitute the coordinates of point C into Newton's Lens Formula to find the value of \(f^2\).
\[ f^2 = x \times y \] \[ f^2 = (20 \, \text{cm}) \times (20 \, \text{cm}) \] \[ f^2 = 400 \, \text{cm}^2 \]Step 4: Calculate the focal length (f) by taking the square root of \(f^2\).
\[ f = \sqrt{400 \, \text{cm}^2} \] \[ f = 20 \, \text{cm} \]The focal length of the convex lens is 20 cm.
Top Questions on Optics
- A thin convex lens of focal length \( 5 \) cm and a thin concave lens of focal length \( 4 \) cm are combined together (without any gap), and this combination has magnification \( m_1 \) when an object is placed \( 10 \) cm before the convex lens.
Keeping the positions of the convex lens and the object undisturbed, a gap of \( 1 \) cm is introduced between the lenses by moving the concave lens away. This leads to a change in magnification of the total lens system to \( m_2 \).
The value of \( \dfrac{m_1}{m_2} \) is
The strain-stress plot for materials A, B, C and D is shown in the figure. Which material has the largest Young's modulus?
- A thin prism with angle 5° of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is :
- A convex lens of refractive index \(1.5\) and focal length \(f=18\) cm is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is \( \alpha \times f \). Find the value of \( \alpha \). (Given: refractive index of water \(=\dfrac{4}{3}\))
- Given below are two statements: Statement I: A plane wave after passing through a prism remains a plane wave, but passing through a small pin hole may become a spherical wave. Statement II: The curvature of a spherical wave emerging from a slit will increase for increasing slit width. In the light of the above statements, choose the correct answer:
Questions Asked in JEE Main exam
Which of the following best represents the temperature versus heat supplied graph for water, in the range of \(-20^\circ\text{C}\) to \(120^\circ\text{C}\)?
- JEE Main - 2026
- Thermodynamics
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid rises upto certain height \( h \). If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by \(1%\) each, then the height of the liquid in the tube will change by ________ %.
- JEE Main - 2026
- thermal properties of matter
- A gas of certain mass filled in a closed cylinder at a pressure of $3.23\,\text{kPa}$ has temperature $50^\circ$C. The gas is now heated to double its temperature. The modified pressure is ___ Pa.
- JEE Main - 2026
- Thermodynamics
- Let the circle \(x^2+y^2=4\) intersect the \(x\)-axis at points \(A(a,0)\) and \(B(b,0)\). Let \(P(2\cos\alpha,2\sin\alpha)\), \(0<\alpha<\frac{\pi}{2}\), and \(Q(2\cos\beta,2\sin\beta)\) be two points on the circle such that \((\alpha-\beta)=\frac{\pi}{2}\). Then the point of intersection of lines \(AQ\) and \(BP\) lies on:
- JEE Main - 2026
- Circles
- The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (\(R \le L\)) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M) :
- JEE Main - 2026
- Kinematics