Nelson Science Perspectives 10
Nelson Science Perspectives 10
1st Edition
Christy C. Hayhoe, Doug Hayhoe, Jeff Major, Maurice DiGiuseppe
ISBN: 9780176355289
Textbook solutions

All Solutions

Section 13-4: The Lens Equations

Exercise 1
Step 1
1 of 3
Data:

$$
d_0=32
$$

$$
f=23
$$

Step 2
2 of 3
Formula:

$$
dfrac{1}{d_0}+dfrac{1}{d_i}=dfrac{1}{f}
$$

Rearrange to solve for $d_i$:

$$
dfrac{1}{d_i}=dfrac{1}{f}-dfrac{1}{d_0}
$$

Simplify using least common denominator:

$$
dfrac{1}{d_i}=dfrac{d_0-f}{fd_0}
$$

Rearrange:

$$
d_i=dfrac{fd_0}{d_0-f}
$$

Substitute values:

$$
d_i=dfrac{23(32)}{32-23}
$$

Simplify and evaluate:

$$
d_i=81.bar7
$$

The image of the frog is $81.bar7$ cm away from the lens.

Result
3 of 3
The image of the frog is $81.bar7$ cm away from the lens.
Exercise 2
Step 1
1 of 3
Data:

$$
d_0=53
$$

$$
d_i=-18
$$

Step 2
2 of 3
Formula:

$$
dfrac{1}{f}=dfrac{1}{d_0}+dfrac{1}{d_i}
$$

Substitute values:

$$
dfrac{1}{f}=dfrac{1}{53}-dfrac{1}{18}
$$

Simplify and evaluate:

$$
dfrac{1}{f}=dfrac{18-53}{53(18)}=-dfrac{35}{954}
$$

Take reciprocal:

$$
f=-dfrac{954}{35}=-27.26
$$

The focal length of the diverging lens is 27.26 cm. The negative sign indicates that it is virtual in nature and on the same side of the lens as the object and its image.

Result
3 of 3
The focal length of the diverging lens is 27.26 cm. The negative sign indicates that it is virtual in nature and on the same side of the lens as the object and its image.
Exercise 3
Step 1
1 of 3
`Data:

$$
d_i=-13
$$

$$
f=34
$$

Step 2
2 of 3
Formula:

$$
dfrac{1}{d_0}+dfrac{1}{d_i}=dfrac{1}{f}
$$

Rearrange to solve for $d_o$:

$$
dfrac{1}{d_o}=dfrac{1}{f}-dfrac{1}{d_i}
$$

Simplify using least common denominator:

$$
dfrac{1}{d_o}=dfrac{d_i-f}{fd_i}
$$

Rearrange:

$$
d_o=dfrac{fd_i}{d_i-f}
$$

Substitute values:

$$
d_o=dfrac{(34)(-13)}{-13-34}
$$

Simplify and evaluate:

$$
d_o=9.404
$$

The object is $9.404$ cm away from the lens.

Result
3 of 3
The object is $9.404$ cm away from the lens.
Exercise 4
Step 1
1 of 3
Data:

$$
d_o=11
$$

$$
f=16
$$

Step 2
2 of 3
Formula:

$$
dfrac{1}{d_0}+dfrac{1}{d_i}=dfrac{1}{f}
$$

Rearrange to solve for $d_i$:

$$
dfrac{1}{d_i}=dfrac{1}{f}-dfrac{1}{d_o}
$$

Simplify using least common denominator:

$$
dfrac{1}{d_i}=dfrac{d_o-f}{fd_o}
$$

Rearrange:

$$
d_i=dfrac{fd_o}{d_o-f}
$$

Substitute values:

$$
d_i=dfrac{(16)(11)}{11-16}
$$

Simplify and evaluate:

$$
d_i=dfrac{176}{-5}=-35.2
$$

The image is formed $35.2$ cm away from the lens on the same side of the lens as the insect. This is indicated by the negative sign of the value of $d_i$. This negative sign also indicates that the image is virtual in nature and upright in attitude.

Result
3 of 3
The image is formed $35.2$ cm away from the lens on the same side of the lens as the insect. This is indicated by the negative sign of the value of $d_i$.
Exercise 5
Step 1
1 of 4
Data:

$$
h_o=12
$$

$$
h_i=35
$$

Step 2
2 of 4
a.

Equation of magnification in terms of height:

$$
M=dfrac{h_i}{h_o}
$$

Substitute values:

$$
M=dfrac{35}{12}
$$

Evaluate:

$$
M=2.9bar1
$$

The magnification of the lens is $2.9bar1$.

Step 3
3 of 4
b. Since the image is formed on the other side of the lens and is inverted in attitude, it can safely be concluded that this is a real image.
Result
4 of 4
a. $M=2.9bar1$

b. Real image

Exercise 6
Step 1
1 of 3
Data:

$$
h_o=14
$$

$$
h_i=7.9
$$

Step 2
2 of 3
Equation of magnification in terms of height:

$$
M=dfrac{h_i}{h_o}
$$

Substitute values:

$$
M=dfrac{7.9}{14}
$$

Evaluate:

$$
M=0.564
$$

The magnification of the lens is $0.564$.

Result
3 of 3
The magnification of this converging lens is $0.564$
Exercise 7
Step 1
1 of 4
Data:

$$
h_o=2.8
$$

$$
h_i=1.3
$$

Step 2
2 of 4
a.

Equation of magnification in terms of height:

$$
M=dfrac{h_i}{h_o}
$$

Substitute values:

$$
M=dfrac{1.3}{2.8}
$$

Evaluate:

$$
M=0.464
$$

The magnification of the lens is 0.464.

Step 3
3 of 4
b. Since the image is formed on the same side of the lens and is virtual, it can safely be concluded that this is upright in attitude.
Result
4 of 4
a. $M=0.464$

b. Image is upright in attitude

Exercise 8
Step 1
1 of 5
Data:

$$
d_o=9.4
$$

$$
M=5.6
$$

Step 2
2 of 5
a.

Equation of magnification in terms of distance:

$$
M=dfrac{d_i}{d_o}
$$

Rearrange to solve for $d_i$:

$$
d_i=Mtimes d_o
$$

Substitute values:

$$
d_i=5.6times 9.4
$$

Evaluate:

$$
d_i=52.64
$$

The image is located at a distance of 52.64 cm from the lens on the same side as the object (fork).

Step 3
3 of 5
b.

Lens equation:

$$
dfrac{1}{f}=dfrac{1}{d_0}+dfrac{1}{d_i}
$$

Substitute values. Note that the image is virtual, therefore $d_i$ is taken to be $-52.64$:

$$
dfrac{1}{f}=dfrac{1}{9.4}+dfrac{1}{-52.64}
$$

Simplify:

$$
dfrac{1}{f}=0.1064-0.0189=0.0875
$$

Evaluate:

$$
f=dfrac{1}{0.0875}=11.429
$$

The focal length of this lens is 11.429 cm.

Step 4
4 of 5
c. Since the virtual image is larger in size than the object, it can safely be concluded that this is a converging lens. Diverging lenses only form virtual images that are smaller in size than the object.
Result
5 of 5
a. Image is located at a distance of 52.64 cm from the lens and on the same side of the lens as the fork.

b. Focal length of the lens is 11.429 cm.

c. This is a converging lens.

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