# Potential Divider Circuits

1. Mar 24, 2017

### Jimmy87

1. The problem statement, all variables and given/known data
Consider a potential divider circuit with an LDR and a fixed resistor. The circuit is powered by a battery which has an emf of 45V with negligible internal resistance.

(a) Describe and sketch a circuit that could use a voltmeter or an ammeter to act as a light meter such that an increase in light intensity will cause an increase in the light meter reading.
(b) The light intensity can cause a variation in the resistance of the LDR from 250-1500 Ohms. If the available options for the fixed resistor are 750 or 1000 Ohms. Use calculations to show which fixed resistor will give the biggest scale change on the light meter.
(c) How would you find the value for the fixed resistor that would give the largest possible scale change on the light meter.
2. Relevant equations
VT/(R1+R2) = Vo/R2 where R1 is the LDR, R2 is the fixed resistor, VT is the emf (45V), Vo is the potential difference across R2.

3. The attempt at a solution
(a) If choosing the ammeter as light meter it simply goes in series with the rest of the components. If choosing voltmeter must go across R2 in order to increase with increasing light intensity.
(b) From my calculations the 750 ohm resistor would give a voltage range of 18.75V across R2 when the LDR varies from 250-1500. The 1000 ohm resistor would give a range of 18V so the 750 ohm resistor is the best choice.
(c) I have no idea with this part! I can randomly select resistances and calculate the range they would give but I have no idea how you find the exact resistance that gives the largest range?

Thanks for any help given.

2. Mar 24, 2017

### Staff: Mentor

Can you write a single expression that would give you the range for a given value of fixed resistor R2?

3. Mar 25, 2017

### Jimmy87

Thanks. Sorry I don't understand if your asking me to write an expression (i.e. it can be done) or whether you are implying it can't be done? If it can, I'm not sure how to do this?

4. Mar 25, 2017

### Staff: Mentor

You should be able to write an expression. You've already done it piece-wise when you found the range for the individual 750 and 1000 Ohm fixed resistors. What steps did you use to do that?

5. Mar 25, 2017

### Jimmy87

I did:

VTR2/(R1+R2) for R1 = 250 and 1500 when R2 = 750 and then R2 = 1000

The only thing I can think of along your lines is:

Vo1-Vo2 = VTR2/(R1+R2) - VTR2/(R3+R2) where R1 is when ldr is at 250 and R3 is when ldr is at 1500 and Vo1 and Vo2 are the output voltages across R2 therefore Vo1 -Vo2 is the range. However, trying to simply this expression makes it look more and more complicated the more I manipulate it :(

Is any of that right?

6. Mar 25, 2017

### Staff: Mentor

You're on the right track and you're nearly there. Let's clean up the math a bit by making the known fixed values (the LDR light and dark resistances) constants and letting the one unknown resistor be R.

$Range = VT \left[ \frac{R}{R + 250} - \frac{R}{R + 1500} \right]$

The source voltage VT just scales the problem since it multiplies everything by a constant value, so you can just concentrate on the terms involving resistor R as they will determine amount of VT that comprises the range. So as a function of R:

$f(R) = \frac{R}{R + 250} - \frac{R}{R + 1500}$

How do go about finding the minimum or maximum of a function?

7. Mar 25, 2017

### Jimmy87

Thanks. That equation makes sense. However, I'm not sure how to find max and min of this function?

8. Mar 25, 2017

### Staff: Mentor

Have you studied calculus? If not you might use a graphical approach.

9. Mar 25, 2017

### Jimmy87

Yes done A fair bit. Done power rule, chain rule and quotient rule. What do I need to use for this problem?

10. Mar 25, 2017

### Staff: Mentor

You should have studied finding the extrema of a function. Say you have a function f(x) and you want to minimize or maximize it. What was the procedure?

11. Mar 25, 2017

### Jimmy87

Ok, so I have looked through all my notes. We did this when we did suvat equations of motion. For example, we used derivatives to find the maximum height a ball can reach. We found the slope of the function by just differentiating the expression for height with respect to time. We then set this derivative equal to zero which would give the time when the slope is zero hence the height is maximum.

So do I differentiate your expression with respect to 'R' then set this derivative equal to zero and see what I get? How do you evaluate the maximum value though?

12. Mar 25, 2017

### Staff: Mentor

Right!
The value(s) of R you obtain from the above procedure will correspond to either maxima or minima for the function. You can either apply the second derivative test (not recommended here since the derivative will get pretty messy), or just plug in values close to but on either side of R to confirm that R gives you a maximum. Or, just plot the original function to show that there's a maximum around the value of R and so R is that maximum.

13. Mar 25, 2017

### Jimmy87

So to differentiate your expression what rule do I use as R is on the numerator and denominator (which is quotient rule) but is also in two separate expressions?

14. Mar 25, 2017

### Staff: Mentor

The quotient rule applies.

15. Mar 25, 2017

### Jimmy87

Ok so don't know if it's correct but I did the quotient rule on each term so I got:

(R+250) - R251/(R + 250)*2 - (R+1500) - R1501/(R+1500)*2

Where *2 is squared.

Have no idea if I have applied quotient rule correctly here?

16. Mar 25, 2017

### Staff: Mentor

The convention is to use "^" to signify exponents. Or you could use the edit panel x2 and x2 buttons to make actual subscripts and superscripts.

Your derivative doesn't look right to me. Maybe take one of the terms and show us your step by step work. Show us the derivative for:

R/(R + 250)

17. Mar 25, 2017

### Jimmy87

Ah I think I made a really silly mistake. Ok so for the above:

For quotient rule U = R and V = R+250.

So du/dR V = R+250
dv/dR U = R
So we get:

R+250 - R / (R+250)^2

Is that right?

18. Mar 25, 2017

### Staff: Mentor

It would be right if you used parentheses to specify the correct order of operations. As it stands the standard interpretation of your expression would be wrong.

(R + 250 - R)/(R + 250)^2

= 250/(R + 250)^2

19. Mar 25, 2017

### Jimmy87

Yes sorry I should have used brackets correctly. So using original expression you gave me the second term differentiated would be:

1500/(R+1500)^2

So the whole expression differentiated is:

250/(R+250)^2 - 1500/(R+1500)^2

So I set this equal to zero and solve for R? If so I get:

R = sqrt (250-1500)

So R=35

Is that right?

20. Mar 25, 2017

Right.