Question

An object 6.0 cm high is placed 30.0 cm from a concave mirror of focal length 10.0cm.

What is the image distance did​i​​. Give your answer with the correct sign convention for mirrors and calculate the size of the image.

Answer 1

The image distance can be found using the mirror formula, which states that 1/f = 1/do + 1/di, where f is the focal length of the mirror, do is the object distance, and di is the image distance. By plugging in the given values, we can solve for di and find that it is -10 cm.

Step-by-step explanation:

To find the image distance using the mirror formula, we use the equation:

1/f = 1/do + 1/di

where f is the focal length of the concave mirror, do is the object distance, and di is the image distance. Plugging in the values from the question, we have:

-1/10 = 1/30 + 1/di

Simplifying the equation gives us -3/30 = 1/di.

Solving for di, we find that the image distance is -10 cm.

Answer 2

The image distance
`(\(d_i\))` for the object placed 30.0 cm from a concave mirror with a focal length of 10.0 cm is calculated using the mirror formula:

`\[ (1)/(f) = (1)/(d_o) + (1)/(d_i) \]`

where
`\(f\)` is the focal length,
`\(d_o\)` is the object distance. Substituting the given values, the image distance is found to be -15.0 cm. The negative sign indicates that the image is formed on the same side as the incident light.

Additionally, to calculate the size of the image (\(h_i\)), we can use the magnification formula:

`\[ M = -(d_i)/(d_o) \]`

Substituting the values, we find the magnification (\(M\)) is -0.5. The negative sign signifies an inverted image.

Step-by-step explanation:

1. Calculation of Image Distance
`(\(d_i\)):`

Using the mirror formula
`\((1)/(f) = (1)/(d_o) + (1)/(d_i)\)`, where
`\(f = 10.0\)` cm and
`\(d_o = -30.0\)` cm, we substitute these values to solve for
`\(d_i\):`

`\[ (1)/(10) = (1)/(-30) + (1)/(d_i) \]`

Solving for
`\(d_i\):`

`\[ (1)/(d_i) = (1)/(10) - (1)/(-30) \]`

`\[ (1)/(d_i) = (1)/(10) + (1)/(30) \]`

`\[ (1)/(d_i) = (4)/(30) \]`

`\[ d_i = (30)/(4) \]`

`\[ d_i = 7.5 \, \text{cm} \]`

The negative sign indicates that the image is formed on the same side as the incident light, characteristic of a virtual image formed by a concave mirror.

2. Calculation of Magnification
`(\(M\))`:

The magnification formula
`\(M = -(d_i)/(d_o)\)` is used. Substituting the known values
`\(d_i = -15.0\)` cm and
`\(d_o = -30.0\)` cm:

`\[ M = -(-15.0)/(-30.0) \]`

`\[ M = -0.5 \]`

The negative sign indicates that the image is inverted compared to the object.

In conclusion, the concave mirror forms a virtual and inverted image with an image distance of -15.0 cm, and the size of the image is half the size of the object. These results align with the conventions and characteristics associated with concave mirrors.