# Minimum angle of resolution - units clarification

1. Aug 14, 2009

### Elemental_

1. The problem statement, all variables and given/known data
A mouse has an pupil diameter of 2.5 mm. lying 200 meters from the mouse are two stones 30 cm apart. λ = 500 nm
a) what angle do the stones subtend?
b) what is the MAR (minimum angle of resolution) of the mouse's eye.

2. Relevant equations
question a) tanθ = O/A
O= 0.3 m
A= 200 m

b) θmin = (1.22*λ )/d
d = pupil diameter

3. The attempt at a solution

for a) I calculated the answer to be 1.5 x 10^-3 however I'm not sure whether the unit for this would be in rads or degrees as the question asks for the angle? I'm aware that MAR (theta-min), the angle between two point sources needed in order for them to be resolved as seperate is measured in radians - does this hold true for the angle that the stones subtend as well?

If the answer i calculated is indeed in the unit of degrees, would i have to convert it to rads? (which would be equivalent to 5.23 x 10^-3 rads)?

b) Minimum angular resolution i calculated to be 2.44 x 10^-4 rads using the formula 1.22(λ)/(diameter of pupil) - I'm pretty sure that this is measured in radians.

If anyone could help me in this it would be greatly appreciated!

2. Aug 14, 2009

### cepheid

Staff Emeritus
Welcome to PF, Elemental_

Yeah, angles are measured in radians. Note, however, that the radian is a *dimensionless* unit. In other words, angles are dimensionless because they are defined as a ratio of two lengths (in the radian system). In part (a), you calculated the angle by dividing a length by a length. The units (metres) cancelled out, leaving you with a dimensionless number. We just assign units to this dimensionless number called radians to indicate that it happens to be an angle.

The only way that the answer you calculated in part (a) could be in degrees would be if you used the wrong mode on your calculator when calculating the arctangent. Be careful to always use angles in radians in your calculations. The formulae you are using rely upon the angles being in radians...those formulae aren't even valid if the angles are not in radians (because they rely upon the definition of an angle as given in the radian system).

3. Aug 14, 2009

### Elemental_

Thank you so much for the thorough explanation cepheid; I verified this by using the degree mode on my calculator instead, and as you said, the answer for a) was a completely different value (although converting this to rads afterwards using pi/180 is a viable option as well, but in my opinion it's not worth it for the extra step). Thanks again for the help!