Answer :
To determine the correct mirror formula for curved mirrors, let's clearly understand the standard relationship between the focal length (f), the object distance (d₀), and the image distance (dᵢ).
For curved mirrors, particularly for spherical mirrors, the mirror formula that relates these three quantities is essential for finding the focal length or determining the position of an image formed by the mirror.
The standard mirror equation is:
[tex]\[ \frac{1}{f} = \frac{1}{d_0} + \frac{1}{d_i} \][/tex]
This equation essentially says that the reciprocal of the focal length of the mirror is equal to the sum of the reciprocals of the object distance and the image distance.
Now let's evaluate each of the given options one by one:
1. [tex]\[ \frac{1}{f}=\left(\frac{1}{d_0}\right)\left(\frac{1}{d_i}\right) \][/tex]
- This option implies that the product of the reciprocals of the object distance and the image distance equals the reciprocal of the focal length, which is incorrect based on the standard mirror formula.
2. [tex]\[ \frac{1}{f}+\frac{1}{d_0}=\frac{1}{d_i} \][/tex]
- This suggests that the sum of the reciprocal of the focal length and the reciprocal of the object distance equals the reciprocal of the image distance, which does not conform to the standard mirror formula.
3. [tex]\[ \frac{1}{f}=\frac{1}{d_0}+\frac{1}{d_i} \][/tex]
- This is the correct formula. It indicates that the reciprocal of the focal length is equal to the sum of the reciprocals of the object distance and the image distance, which matches the standard mirror equation.
4. [tex]\[ \frac{1}{f}-\frac{1}{d_0}-\frac{1}{d_j} \][/tex]
- This option suggests subtracting the reciprocals, which is not the correct relationship as defined by the standard mirror formula.
Thus, the correct mirror formula for curved mirrors from the given options is:
[tex]\[ \frac{1}{f}=\frac{1}{d_0}+\frac{1}{d_i} \][/tex]
Therefore, the correct answer is:
[tex]\[\boxed{3}\][/tex]
For curved mirrors, particularly for spherical mirrors, the mirror formula that relates these three quantities is essential for finding the focal length or determining the position of an image formed by the mirror.
The standard mirror equation is:
[tex]\[ \frac{1}{f} = \frac{1}{d_0} + \frac{1}{d_i} \][/tex]
This equation essentially says that the reciprocal of the focal length of the mirror is equal to the sum of the reciprocals of the object distance and the image distance.
Now let's evaluate each of the given options one by one:
1. [tex]\[ \frac{1}{f}=\left(\frac{1}{d_0}\right)\left(\frac{1}{d_i}\right) \][/tex]
- This option implies that the product of the reciprocals of the object distance and the image distance equals the reciprocal of the focal length, which is incorrect based on the standard mirror formula.
2. [tex]\[ \frac{1}{f}+\frac{1}{d_0}=\frac{1}{d_i} \][/tex]
- This suggests that the sum of the reciprocal of the focal length and the reciprocal of the object distance equals the reciprocal of the image distance, which does not conform to the standard mirror formula.
3. [tex]\[ \frac{1}{f}=\frac{1}{d_0}+\frac{1}{d_i} \][/tex]
- This is the correct formula. It indicates that the reciprocal of the focal length is equal to the sum of the reciprocals of the object distance and the image distance, which matches the standard mirror equation.
4. [tex]\[ \frac{1}{f}-\frac{1}{d_0}-\frac{1}{d_j} \][/tex]
- This option suggests subtracting the reciprocals, which is not the correct relationship as defined by the standard mirror formula.
Thus, the correct mirror formula for curved mirrors from the given options is:
[tex]\[ \frac{1}{f}=\frac{1}{d_0}+\frac{1}{d_i} \][/tex]
Therefore, the correct answer is:
[tex]\[\boxed{3}\][/tex]
