Minimum Luminosity between Two Light Sources

[guitarguy]

I am having trouble with the following problem.

"The illumination from a light source is directly proportional to the stength of the source and inversly proportional to the square of the distance from the source. If two light sources of strengths S1 and S2 are d units apart, at what point on the line segment joining the two sources is the illumination minimal?

I understand I should write an equation for the lumninosity of each source in terms of d, take the derivative set to zero and find the minimum.

I called the distance between the sources x. And, the point of minimum as d.

Then Illumination of source one: I1=(K*S1)/d^2

The illumination of source two: I2= (K*S2)/(x-d)^2

I added I1 plus I2 to get total luminosity contributed by both sources. Took the derivative but this is wrong?

I have tried problem many, quite stuck.

Thank you

 

[tkhunny]

I called the distance between the sources x. And, the point of minimum as d.

Then Illumination of source one: I1=(K*S1)/d^2

This is very strange. You are given in the problem statement that the light sources are 'd' apart. Why did you go to the trouble to redefine this to "x" and then use "d" for something else?

What was your derivative? /dx or /dd?

You may have to decide S1 > S2 or the other way around. Why?

 

[HallsofIvy]

Set up a coordinate system so that S1 is at x= 0 and S2 is at x= d. The illumination at x is \(\displaystyle kS1/x^2+ kS2/(d- x)^2= k(S1x^{-2}+ S2(d- x)^{-2})\). Differentiate that with respect to x and set the derivative equal to 0.