Coils arrangement so the hotplate operates at 3 diff. powers

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1. Oct 8, 2016

moenste

1. The problem statement, all variables and given/known data
A certain electric hotplate, designed to operate on a 250 V supply, has two coils of nichrome wire of resistivity 9.8 * 10-7 Ω m. Each coil consists of 16 m of wire of cross-sectional area 0.20 mm2.

(a) For one of the coils calculate (i) its resistance, (ii) the power dissipation when 250 V supply is connected across the coil, assuming its resistance does not change with temperature.

(b) Show, by means of diagrams, how these coils may be arranged so that the hotplate may be made to operate at three different powers. In each case, calculate the power rating.

(c) The hotplate is connected to the 250 V supply by means of cable of total resistance 3.0 Ω. (i) Calculate the power loss in the connecting cable when the hotplate is being used on its middle power rating. (ii) Comment qualitatively on any change in power loss in the cable when the hotplate is operating at each of its other power ratings.

(d) Different connecting cables are available for use with the hotplate. The maximum safe current which can be used in any one of the cables is 1 A or 3 A or 6 A or 12 A. State which is the most appropriate cable to use and briefly explain one possible danger of using cable with a lower maximum safe current.

Answers: (a) (i) 78 Ω, (ii) 8 * 102 W, (c) (i) 28 W.

2. The attempt at a solution
(a) (i) ρ = (A R) / L → R = (ρ L) / A = (9.8 * 10 -7 * 16) / (2 * 10-7) = 78.4 Ω.

(a) (ii) P = V2 / R = 2502 / 78.4 = 797 W.

(b) Not even sure where to start. Am I required to make a circuit? What are coils in a circuit? Resistors? What is a hotplate in a circuit? A cell?

(c) (i) As I understand we'll have RTotal = RCoil + RCable = 78 Ω + 3 Ω = 81 Ω. Then P = 2502 / 81 = 771.6 W. Difference between 800 W and 771.6 W is 28.4 W.

(c) (ii) Not sure on this question either. So we need to say whether there will be any power losses in the cable when the hotplate is operating at what? What does "each of its other power ratings" actually mean?

(d) Maybe the higher is the maximum safe current the better it is? So in case more current starts going through the circuit the cables will not overheat and start buring? Just a guess, also not sure on this part.

2. Oct 8, 2016

Staff: Mentor

The coils act as resistors, right. The diagrams mentioned in the problem statement are circuit diagrams. The heating pad is irrelevant - just the electrical part matters.

You need (b) here (e.g.: what about the other coil?), but you picked the right circuit. Why did you calculate the difference to 800 W? The coil cannot have 800 W, that would be more than the power in wire plus coil, and it would give a negative power loss in the cable.
Your answer is not far away from the right answer by accident here, but it is not right.
That needs an answer to (b). Depending on how you connect the coils, you get different heating power values.
Well, the 12 A cable is fine, but more expensive than others. Can you also use cables rated for lower currents?
Overheating is indeed the issue if the current rating is too low.

3. Oct 8, 2016

Staff: Mentor

Yes, you're being asked to draw three configurations where one or more of the coils is being supplied power.

For purposes of this exercise the coils can be represented as resistors.

[edit: I see that @mfb got there first]

4. Oct 8, 2016

moenste

Let's then finish (b) first.

Maybe it should be something like this:

5. Oct 8, 2016

Staff: Mentor

There is no circuit element "hotplate". A hotplate is just a metal disk with heating coils below (or inside). The interesting part as far as this problem is concerned is the two resistive coils. So your circuit diagrams should contain only power supply and resistors.

You need to determine how to apply voltage to the resistors to get three different power settings.

6. Oct 8, 2016

moenste

I can only think of: (i) power supply in series with two resistors in series and (ii) power supply in series with two resistors that are in parallel.

7. Oct 8, 2016

Staff: Mentor

Nowhere does it say that both coils have to be in use at the same time

8. Oct 8, 2016

moenste

Maybe then:
and (iii) power supply in series with just a one resistor.

9. Oct 8, 2016

Staff: Mentor

Yes.

10. Oct 8, 2016

moenste

And this means
that we need to calculate the power (like P = V I or P = V2 / R) for each of the graphs?

11. Oct 8, 2016

Staff: Mentor

Yes.

12. Oct 8, 2016

moenste

Then we would have:
1. One resistor: P = 2502 / 78 = 801 W.
2. Two resistors in series: P = 2502 / (78 + 78) = 401 W.
3. Two parallel resistors: P = 2502 / 39 = 1603 W. (1 / R = 1 / 78 + 1 / 78 → R = 39 Ω.)

13. Oct 8, 2016

Staff: Mentor

Looks good.

14. Oct 8, 2016

moenste

OK, so for the next part we have the middle power 801 W or (797 W if we use 78.4 Ω). So, roughly 800 W.

Then the power loss will be:
What's wrong with this one?

15. Oct 8, 2016

Staff: Mentor

I don't see anything wrong, per se. Does it answer the question posed?

16. Oct 8, 2016

moenste

I would say it's correct, but:
Maybe @mfb meant that I got the right answer by accident withouth considering (b) first?

17. Oct 8, 2016

Staff: Mentor

I think it's a matter of interpretation as to what "power lost" means. Lost as compared to what benchmark?

If the cable was ideal (no resistance) then the single coil produces about 800 W of heat. What heat does that coil produce when the cable is not ideal but has a resistance of 3 Ohms? The difference between "ideal" and "not ideal" could be considered power lost.

On the other hand, you could simply calculate the power consumed by the cable directly under operating conditions and call that the power lost (since it doesn't contribute to heating up your soup).

I think that both approaches should lead to similar, but not identical answers. And I also suspect that the latter approach s what is intended in the question: find the power dissipated by the cable.

18. Oct 8, 2016

moenste

I did that actually. But P = 2502 / 3 = 20 833 W.

While if we do 2502 / (78 + 3) = 771.6 W. 800 - 771.6 = 28.4 W.

19. Oct 8, 2016

Staff: Mentor

But the 250 V is not directly across the cable resistance! It and the hotplate resistance are in series. Better to first find the current, then apply $P = I^2 R$ for the resistances in a series connection.
Yes, but then you need to justify the 800 W value (as I did in post #17). It's easier to just calculate the actual power lost during operation, as above.

20. Oct 8, 2016

Staff: Mentor

I think the problem statement is unambiguous, it asks for the power dissipation in the cable. If it would ask about power differences in the hot plate, it would be "due to the connecting cable" or something like that.