Sherlock Holmes examines a clue by holding

his magnifying glass (with a focal length of
17.0 cm) 12.4 cm away from an object.
Find the image distance.
Answer in units of cm

Find the magnification.

Well, Sherlock Holmes certainly knows how to keep things in focus! To find the image distance, we can use the lens formula:

1/f = 1/do + 1/di

Where f is the focal length, do is the object distance, and di is the image distance. In this case, f = 17.0 cm and do = 12.4 cm.

Plugging these values into the formula, we can rearrange it to solve for di:

1/di = 1/f - 1/do

1/di = 1/17.0 - 1/12.4

Now, let's calculate the right-hand side of the equation:

1/di ≈ 0.0588 - 0.0806

1/di ≈ -0.0218

To find di, we can take the reciprocal:

di ≈ -1 / (-0.0218)

di ≈ 45.87 cm

So, it seems like Sherlock Holmes has found an image distance of approximately 45.87 cm!

Now, let's move on to find the magnification. The magnification (M) is defined as the ratio of the image height (hi) to the object height (ho):

M = hi / ho

But since we don't have any information about the heights, we'll instead use the magnification formula for lenses:

M = - di / do

Plugging in the values we know, we can find the magnification:

M = - 45.87 cm / 12.4 cm

M ≈ -3.7

So, Sherlock Holmes discovered a magnification of roughly -3.7. Just remember, though, that it's always good to have a positive outlook even if the magnification is negative!

To find the image distance, we can use the lens formula:

1/f = 1/d₀ + 1/dᵢ

Where:
- f is the focal length of the lens (17.0 cm)
- d₀ is the object distance (12.4 cm)
- dᵢ is the image distance (what we want to find)

Plugging in the given values, we have:

1/17.0 = 1/12.4 + 1/dᵢ

To solve for dᵢ, we can rearrange the equation:

1/dᵢ = 1/17.0 - 1/12.4

Finding the least common denominator and simplifying, we get:

1/dᵢ = (12.4 - 17.0)/(17.0 * 12.4)
= -4.6 / 210.8

Taking the reciprocal of both sides, we have:

dᵢ = 210.8 / -4.6

Calculating this expression, we find:

dᵢ ≈ -45.83 cm

However, this value is negative, indicating that the image formed is virtual. Since virtual images cannot be physically measured, we conclude that there is no image formed by the magnifying glass at this object distance.

Now let's find the magnification:

The magnification (m) is defined as the ratio of the height of the image (hᵢ) to the height of the object (h₀):

m = hᵢ / h₀

Since no image is formed, the magnification is 0.

To find the image distance, we can use the lens equation, which relates the object distance (distance of the object from the lens, denoted by "do"), the image distance (distance of the image from the lens, denoted by "di"), and the focal length of the lens (denoted by "f"). The lens equation is stated as:

1/f = 1/do + 1/di

In this case, the focal length (f) is given as 17.0 cm, and the object distance (do) is given as 12.4 cm. We are to find the image distance (di). We can rearrange the lens equation to solve for di:

1/di = 1/f - 1/do

Plugging in the values we have:

1/di = 1/17 - 1/12.4

Now, we can calculate the value of 1/di:

1/di = (12.4 - 17) / (17 * 12.4)
= -0.3241 / 209.6
= -0.001547

To get the actual value of di, we take the reciprocal of -0.001547:

di ≈ -1 / (-0.001547)
di ≈ 647.09 cm

Since the value is positive, it means the image is formed on the opposite side of the object compared to the lens. Therefore, the image distance is approximately 647.09 cm.

Now, to find the magnification (M), we can use the formula:

M = -di / do

Plugging in the known values:

M = -647.09 / 12.4
M ≈ -52.30

The negative sign in the magnification indicates that the image formed is inverted compared to the object. Therefore, the magnification is approximately -52.30.