Convex and Plane Mirror Coinciding Image Distance Calculation
This problem involves finding the distance between a convex mirror and a plane mirror when the images formed by both coincide. We are given:
- Object distance from convex mirror, $u = -30$ cm (real object).
- Focal length of convex mirror, $f = +30$ cm (convex mirror).
Image Formed by Convex Mirror
We use the mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ to find the position of the image ($I_1$) formed by the convex mirror.
Substituting the given values:
$ \frac{1}{v} + \frac{1}{-30 \text{ cm}} = \frac{1}{30 \text{ cm}} $ $ \frac{1}{v} - \frac{1}{30 \text{ cm}} = \frac{1}{30 \text{ cm}} $ $ \frac{1}{v} = \frac{1}{30 \text{ cm}} + \frac{1}{30 \text{ cm}} $ $ \frac{1}{v} = \frac{2}{30 \text{ cm}} = \frac{1}{15 \text{ cm}} $ $ v = +15 \text{ cm} $The image $I_1$ is formed 15 cm behind the convex mirror. This is a virtual image.
Plane Mirror Placement and Coinciding Images
A plane mirror is placed at a distance $d$ from the convex mirror. The virtual image $I_1$ formed by the convex mirror acts as the object ($O_2$) for the plane mirror. The final image ($I_2$) formed by the plane mirror must coincide with $I_1$. This means $I_2$ must also be located 15 cm behind the convex mirror.
Let's consider the distance $d$:
- Case 1: Plane mirror is between the convex mirror and $I_1$ ($0 < d < 15$ cm).
- The distance of $I_1$ from the plane mirror is $15 - d$. This acts as the object distance ($O_2$) for the plane mirror.
- For a plane mirror, the image distance ($I_2$) is equal to the object distance ($O_2$). So, $I_2$ is formed $15 - d$ behind the plane mirror.
- The position of $I_2$ relative to the convex mirror is $d + (15 - d) = 15$ cm.
- This location (15 cm) matches the location of $I_1$. Thus, the images coincide for any $d$ in the range $0 < d < 15$ cm.
- Case 2: Plane mirror is beyond $I_1$ ($d > 15$ cm).
- The distance of $I_1$ from the plane mirror is $d - 15$. This acts as the object distance ($O_2$).
- The image $I_2$ is formed $d - 15$ behind the plane mirror.
- The position of $I_2$ relative to the convex mirror is $d + (d - 15) = 2d - 15$ cm.
- For coincidence, $I_2$ must be at 15 cm: $2d - 15 = 15 \implies 2d = 30 \implies d = 15$ cm. This contradicts the condition $d > 15$ cm.
- Case 3: Plane mirror is exactly at $I_1$'s location ($d = 15$ cm).
- The object distance for the plane mirror is $15 - 15 = 0$.
- The image $I_2$ forms at the plane mirror itself.
- Since $I_1$ is also at 15 cm (the location of the plane mirror), the images coincide.
From the analysis, the images coincide for $0 < d \leq 15$ cm.
Determining the Specific Distance
Since multiple values of $d$ satisfy the coincidence condition, we infer a standard interpretation for such problems: the plane mirror is placed midway between the convex mirror and the location of the virtual image formed by the convex mirror.
Position of $I_1 = 15$ cm.
The distance $d$ is half of this position:
$ d = \frac{15 \text{ cm}}{2} = 7.5 \text{ cm} $This distance $d=7.5$ cm falls within the valid range ($0 < d < 15$ cm), confirming that the images coincide.
Final Answer
The distance between the two mirrors is 7.5 cm.
Similar Questions
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure. Distance of the image of an object 'O', is :
(C is the center of curvature of the spherical surface and R is the radius of curvature)- If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is $30^\circ$ in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _________ $\mu$m.
- Two polarisers $P_1$ and $P_2$ are placed in such a way that the intensity of the transmitted light will be zero. A third polariser $P_3$ is inserted in between $P_1$ and $P_2$, at particular angle between $P_2$ and $P_3$. The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers $P_2$ and $P_3$ is :
- In a Young's double slit experiment, two slits are located 1.5 mm apart. The distance of screen from slits is 2 m and the wavelength of the source is 400 nm. If the 20 maxima of the double slit pattern are contained within the central maximum of the single slit diffraction pattern, then the width of each slit is $x\times10^{-3}$ cm, where x-value is ________.
- Two thin convex lenses of focal lengths $30 \text{ cm}$ and $10 \text{ cm}$ are placed coaxially, $10 \text{ cm}$ apart. Thepower of this combination is:
Important Questions from Optics
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
