A typical red blood cell subtends an angle of only 1.7×10−5 rad when viewed at a person's near-point distance of 25 cm. Suppose a red blood cell is examined with a compound microscope in which the objective and eyepiece are separated by a distance of 13.0 cm .
Given that the focal length of the eyepiece is 2.8 cm , and the focal length of the objective is 0.41 cm , find the magnitude of the angle subtended by the red blood cell when viewed through this microscope.
Ah, the fascinating world of blood cells and microscopes! Let me calculate that for you with a touch of humor.
First, let's break out our math hats. We have some numbers to play with here: the objective focal length is 0.41 cm, the eyepiece focal length is 2.8 cm, and they are 13.0 cm apart.
Now, we know that the combined magnification of a compound microscope is the product of the magnification of the objective and the eyepiece. But how does that translate into the angle subtended by the red blood cell?
To find that out, we need to use some trigonometry. We can use the formula:
tanθ = (height of object) / (distance from the object)
In this case, the height of the object (our red blood cell) is not given, but we can assume it is the same for both the actual viewing distance (25 cm) and the magnified viewing distance (13.0 cm).
So, let's plug in the numbers and solve for the angle θ:
tanθ_actual = (height of object) / (25 cm)
tanθ_magnified = (height of object) / (13.0 cm)
Now, we can use the fact that the magnification M is equal to the ratio of the angle subtended by the object in the image to the angle subtended by the object in reality:
M = θ_magnified / θ_actual
So, rearranging the equation, we can find θ_magnified:
θ_magnified = M * θ_actual
Now, we just need to plug in the numbers for M and θ_actual and calculate θ_magnified.
I hope all these calculations didn't make your head spin like a red blood cell in a microscope! Let's do the math and find out the final answer for you.
To find the magnitude of the angle subtended by the red blood cell when viewed through the microscope, we can use the formula for angular magnification:
M = θ/θ₀
Where:
M is the angular magnification
θ is the angle subtended by the object when viewed through the microscope
θ₀ is the angle subtended by the object when viewed without the microscope
We are given:
θ₀ = 1.7×10^(-5) rad
Let's calculate the magnification due to the objective lens:
M_objective = -f_objective / D
Where:
- f_objective is the focal length of the objective lens
- D is the distance between the objective lens and the near-point distance
Given:
- f_objective = 0.41 cm = 0.0041 m
- D = 25 cm = 0.25 m
M_objective = (-0.0041 m) / (0.25 m) = -0.0164
Now, let's calculate the magnification due to the eyepiece:
M_eyepiece = -1 / f_eyepiece
Where:
- f_eyepiece is the focal length of the eyepiece
Given:
- f_eyepiece = 2.8 cm = 0.028 m
M_eyepiece = -1 / (0.028 m) = -35.71
Next, we can calculate the total angular magnification:
M_total = M_objective * M_eyepiece
M_total = (-0.0164) * (-35.71) = 0.5856
Finally, we can calculate the angle subtended by the red blood cell through this microscope:
θ = M_total * θ₀
θ = (0.5856) * (1.7×10^(-5) rad) = 9.947×10^(-6) rad
Therefore, the magnitude of the angle subtended by the red blood cell when viewed through this microscope is approximately 9.947×10^(-6) rad.