Complete Question
The distance between the objective and eyepiece lenses in a microscope is 19 cm . The objective lens has a focal length of 5.5 mm .
What eyepiece focal length will give the microscope an overall angular magnification of 300?
Answer:
The eyepiece focal length is
Explanation:
From the question we are told that
The focal length is 
This negative sign shows the the microscope is diverging light
The angular magnification is 
The distance between the objective and the eyepieces lenses is 
Generally the magnification is mathematically represented as
![m = [\frac{Z - f_e }{f_e}] [\frac{0.25}{f_0} ]](https://tex.z-dn.net/?f=m%20%20%3D%20%20%5B%5Cfrac%7BZ%20-%20f_e%20%7D%7Bf_e%7D%5D%20%5B%5Cfrac%7B0.25%7D%7Bf_0%7D%20%5D)
Where
is the eyepiece focal length of the microscope
Now making
the subject of the formula
![f_e = \frac{Z}{1 - [\frac{M * f_o }{0.25}] }](https://tex.z-dn.net/?f=f_e%20%20%3D%20%5Cfrac%7BZ%7D%7B1%20-%20%5B%5Cfrac%7BM%20%20%2A%20%20f_o%20%7D%7B0.25%7D%5D%20%7D)
substituting values
![f_e = \frac{ 0.19 }{1 - [\frac{300 * -0.0055 }{0.25}] }](https://tex.z-dn.net/?f=f_e%20%20%3D%20%5Cfrac%7B%200.19%20%7D%7B1%20-%20%5B%5Cfrac%7B300%20%20%2A%20%20-0.0055%20%7D%7B0.25%7D%5D%20%7D)
The trick here is that there are two variables, Diameter of the coil and number of turns in it. So, in order to solve it, we will need two equations in D and N
Since we are making coil, formula for magnetic field will be B = μₒni/D
D = μₒni/B = 4π x 10^(-7) x (N) x (2.2) / (1.6)x10^(-3) = 1.7 x 10^(-3) x N = D
Another equation will be from the wire's length.
Total length = Circumference of one loop x No. of turns = πD x N = 2.2
We will eliminate N because we need D.
N = 2.2 / πD
Substitute in 1st equation.
1.7 x 10^(-3) x 2.2 / π = D^2
<span>
D = 0.034 m = 3.4 cm </span>