A compound microscope has lenses of focal length 1.0cm and 5.0cm from the objective. if the final image is formed at the least distance of distinct vision from the eyepiece, calculate i. the separation of the objective and the eyepiece. ii. the angular magnification.
To calculate the separation of the objective and the eyepiece, we need to use the lens formula:
1/f = 1/v - 1/u
where:
f is the focal length of the lens,
v is the image distance, and
u is the object distance.
For the objective lens:
f_objective = 1.0 cm
For the final image formed at the least distance of distinct vision from the eyepiece, v_objective = least distance of distinct vision (d) = 25 cm.
Plugging in the values, we have:
1/1.0 = 1/25 - 1/u_objective
To solve for u_objective, rearrange the equation:
1/u_objective = 1/25 - 1/1.0
1/u_objective = 1/25 - 1/1.0
Now, let's solve for u_objective:
1/u_objective = 1/25 - 1/1.0
1/u_objective = 1/25 - 1/1.0
Simplifying the equation:
1/u_objective = 0.04 - 1
1/u_objective = -0.96
Taking the reciprocal of both sides:
u_objective = -1/(0.96)
u_objective = -1.042 cm
The object distance, u_objective, is negative because it is on the same side as the object.
Now, let's calculate the separation of the objective and the eyepiece:
Separation = u_eyepiece - v_objective
Since the final image formed at the least distance of distinct vision from the eyepiece, v_eyepiece = least distance of distinct vision (d) = 25 cm.
To find u_eyepiece, we'll use the lens formula again.
For the eyepiece lens:
f_eyepiece = 5.0 cm
v_eyepiece = 25 cm
Plugging in the values, we have:
1/5.0 = 1/25 - 1/u_eyepiece
1/5.0 = 1/25 - 1/u_eyepiece
To solve for u_eyepiece, rearrange the equation:
1/u_eyepiece = 1/25 - 1/5.0
1/u_eyepiece = 1/25 - 1/5.0
Now, let's solve for u_eyepiece:
1/u_eyepiece = 0.04 - 1/5.0
1/u_eyepiece = 0.04 - 0.2
Simplifying the equation:
1/u_eyepiece = -0.16
Taking the reciprocal of both sides:
u_eyepiece = -1/0.16
u_eyepiece = -6.25 cm
The object distance, u_eyepiece, is negative because it is on the same side as the object.
Now, let's calculate the separation:
Separation = u_eyepiece - v_objective
Separation = -6.25 cm - 25 cm
Separation = -31.25 cm
The separation of the objective and the eyepiece is -31.25 cm.
ii. To calculate the angular magnification, we use the formula:
Angular Magnification (M) = (Angular Magnification of Objective) x (Angular Magnification of Eyepiece)
Angular Magnification (M) = (Height of Image by Objective) / (Height of Object) x (Height of Image by Eyepiece) / (Height of Image by Objective)
Since the final image is formed at the least distance of distinct vision from the eyepiece, the height of the final image by the eyepiece is equal to the least distance of distinct vision (d) = 25 cm.
To calculate the height of the image by the objective lens, we use the magnification formula:
Height of Image by Objective / Height of Object = v_objective / u_objective
Plugging in the values, we have:
Height of Image by Objective / Height of Object = 25 cm / -1.042 cm
Simplifying the equation:
Height of Image by Objective / Height of Object ≈ -24 cm / cm ≈ -24
The negative sign indicates that the image formed by the objective lens is inverted.
Now, let's calculate the angular magnification:
Angular Magnification (M) = (-24 cm / cm) x (25 cm / 25 cm)
Angular Magnification (M) = -24 x 1
Angular Magnification (M) = -24
The angular magnification is -24.
To solve this problem, we can use the lens formula and the formula for angular magnification.
i. The separation of the objective and the eyepiece:
The compound microscope consists of two lenses - the objective lens and the eyepiece lens. Let's denote the separation between these lenses as "d" (in cm).
Using the lens formula, we have:
1/f_obj = 1/d_o - 1/do
where f_obj is the focal length of the objective lens, and do is the distance of the object from the objective lens.
Given that the focal length of the objective lens is 1.0 cm, and the distance between the objective and final image (formed at the least distance of distinct vision) is the distance of the object from the objective, we can rewrite the lens formula as:
1/1.0 = 1/d_o - 1/do
Simplifying this equation, we get:
1 = (do - d_o) / (d_o * do)
The distance of the object from the objective lens (do) can be taken as the distance between the objective and the eyepiece (d):
1 = (d - 1.0) / (1.0 * d)
Cross-multiplying, we have:
d = 2.0 cm
Therefore, the separation of the objective and the eyepiece is 2.0 cm.
ii. The angular magnification:
The formula for angular magnification (M) of a compound microscope is given as:
M = (1 + (d_o / f_obj)) * (1 / (1 - d_e / f_e))
where d_o is the distance between the eyepiece and the final image, f_obj is the focal length of the objective lens, d_e is the distance between the eyepiece and the observer's eye, and f_e is the focal length of the eyepiece lens.
Given that the final image is formed at the least distance of distinct vision, the distance between the eyepiece and the observer's eye (d_e) can be taken as 25 cm (standard near point or least distance of distinct vision).
Also, the focal length of the eyepiece lens (f_e) is given as 5.0 cm.
Using these values in the formula for angular magnification, we have:
M = (1 + (d/1.0)) * (1 / (1 - 25/5.0))
Substituting the value of d (separation of the objective and the eyepiece) as 2.0 cm, we get:
M = (1 + (2.0/1.0)) * (1 / (1 - 25/5.0))
Simplifying the equation, we have:
M = 3 * (1 / (1 - 5/5.0))
M = 3 * (1 / (1 - 1.0))
M = 3 * (1 / 0)
M = 3 * ∞
Since 1 / 0 is undefined, the angular magnification (M) is infinity (∞).
Therefore, the separation of the objective and the eyepiece is 2.0 cm, and the angular magnification is infinity.