Question

Two plane mirrors are placed parallel to each other and $\mathbf{40}\mathbf{}\mathbf{\text{cm}}$apart. An object is placed $\mathbf{10}\mathbf{}\mathbf{\text{cm}}$from one mirror. Determine the (a) smallest, (b) second smallest, (c) third smallest (occurs twice), and (d) fourth smallest distance between the object and image of the object.

Short answer

1. The smallest distance between the object and the image of the object is $20\text{cm}$.
2. The second smallest distance between the object and the image of the object is$60\text{cm}$.
3. The third smallest (occurs twice) distance between the object and the image of the object is$80\text{cm}$.
4. The fourth smallest distance between the object and the image of the object is $100\text{cm}$.

Step-by-step solution

The given data:

• Distance between two parallel plane mirrors is$40\text{cm}$.
• Distance between object and first mirror is $10\text{cm}$.

Understanding the concept of image formation in theplane mirror

As two plane mirrors are placed parallel to each other separated by a distance of $\mathbf{40}\mathbf{\text{cm}}$, they will form images as shown in the below diagram. Each mirror produces a first image of the object. Then, each mirror produces a second image with the object being the first image in the opposite mirror. Then each mirror produces a third image with the object being the second image in the opposite mirror, and so on. Therefore, the mirrors will produce five images. Here, you find the smallest distance, second smallest distance, third smallest distance, and fourth smallest distance between the object and an image of the object.

(a) Calculation of the smallest distance between the object and the image:

To know the smallest distance, yourefer the above figure. In the closest mirror$M_1$, the “first” image$I_1$is $10\text{cm}$ behindrole="math" localid="1663022915939" $M_1$and thus,$20\text{cm}$ from the object O. This is the smallest distance between the object and an image of the object.

Hence, the smallest distance is $20\text{cm}$.

(b) Calculation of the second smallest distance between the object and the image:

To know the second smallest distance, refer the above figure. There are images from both Oand$I_1$in the more distant mirror,$M_2$an image$I_2$located at $30\text{cm}$behind$M_2$. Since Ois $30\text{cm}$ in front of it,$I_2$is $60\text{cm}$ from O. This is the second smallest distance between the object and an image of the object.

Hence, the second smallest distance is $60\text{cm}$.

(c) Calculation of the third smallest distance between the object and the image:

To know the third smallest distance, refer the above figure. There is also an image $I_3$that is $50\text{cm}$behind $M_2$ (since $I_1$ is $50\text{cm}$ in front of it). Thus, $I_3$is $80\text{cm}$from O. In addition, you have another image $I_4$that is $70\text{cm}$ behind role="math" localid="1663023685109" $M_1$ (since $I_2$ is $70\text{cm}$ in front of it). The distance from $I_4$ to O for is $80\text{cm}$.

Hence, the third smallest distance is $80\text{cm}$.

(d) Calculation of the fourth smallest distance between the object and the image

To know the fourth smallest distance, yourefer the above figure. Returning to the closer mirror$M_1$, there is an image$I_5$that is $90\text{cm}$ behind the mirror (sincerole="math" localid="1663023840894" $I_3$is $90\text{cm}$ in front of it). The distance (measured from O) is found to be $100\text{cm}$for$I_5$.

Hence, the fourth smallest distance is $100\text{cm}$.