The reason why nichrome wire heat up is usually explained by the fact that it has high resistance. While I don’t doubt this explanation, I can’t get my head around the following “paradox.”

If the current is known, then the power generated by a heating element is proportional to the resistance:

$$P = I^2 R$$

Usual power sources are fixed voltage (not fixed current) hence the power is inversely proportional to R:

$$P = \frac{U^2}{R}$$

The “paradox” is in that the latter seems to tell us that we need less resistance to generate more heat (so that copper is better than nichrome).

One could try to take into account the resistance of the conducting (say) copper wire, but it doesn’t seem to help.

• Consider Ohm's Law and algebraic substitution. Apr 30, 2023 at 9:33
• Think cost and practically. If you want to make an X W heater running of Y V supply, how much copper would it take? How long and thin is it? Now do the same for nichrome. Apr 30, 2023 at 10:57
• You are mixing several concepts. The main reasons to use Nichrome are - | 1. It has a relatively stable resistance with temperature variation.| 2. It tends not to oxidise rapidly at high temperatures. | 3. It has a highish melting point | 4. It's high resistivity means that you can produce resistors of required resistance and power ratings with RELATIVELY small amounts of wire. || Copper is poor in all these areas. Iron is not much better than copper. Apr 30, 2023 at 11:20
• Switch from resistance to resistivity so that you can explicitly see the dependence from geometry. And then ask yourself, do I want a very long very thin wire for my application, or do I want a short and thicker one? Which one is better in terms of space occupied, strength, and temperature reached at equilibrium with the surroundings? Apr 30, 2023 at 13:26

The statement

The reason why nichrome wire heat up is usually explained by the fact that it has high resistance.

is problematic because it contains a true statement but is misleading. A better version of this may read

"One reason that Nichrome is suitable for heating elements is that it's high resistivity reduces the amount of material required to manufacture a resistor of given resistance and power rating".

This tends to get shortened into what is often stated.

For a given electrical heating task the applied voltage and required power between them define the required resistance.
ie R = V^2/P

Notionally the resistance could be provided by a wide range of materials. eg Wire made from iron, copper, gold, silver, nichrome, ... . Or from Yak's butter, or carbon resistive ink or much else.
The unsuitability of some of these materials is obvious (Yak's butter, gold, ...) but others are less obviously unsuitable. As further discussed below, copper wire is seldom the best choice because it tends to melt, has a low resistivity so a large amount (relatively) is required, it changes resistance with temperature, and it oxidises quite rapidly in air. Nichrome is relatively better in all these areas.

The main reasons to use Nichrome are

1. It has a relatively stable resistance with temperature variation.
Temperature coefficient of resistance per degree Celsius ~=
Copper ....... 0.4%
Iron .............. 0.6%
Nichrome .. 0.015%
1. It tends not to oxidise rapidly at high temperatures.
1. It has a highish melting point.
Nichrome ~= 1400 degrees Celsius. Copper ~= 1000 degree C. Iron ~= 1500 C.
1. It's high resistivity means that you can produce resistors of required resistance and power ratings with RELATIVELY small amounts of wire.
Copper ~ 1.7 E-8 Ohm meter
Iron 9.7 Ohm-metre
Nichrome 100 Ohm-metre

Copper is poor in all these areas.
When used as a heating element exposed to air it tends to change resistance substantially as it heats, oxidise rapidly, melt and require long and thin wire for most applications.

Iron is not much better than copper.

For eg a 1 kW heating element operating at 100 VAC the required resistance is R = P/V = 10 Ohms.
The required element is made thick enough to not melt (required thickness depends whether air would or in contact with a thermal sink) and then long enough to produce the required resistance.

Table of resistivities here

Table of melting points here

Table of temperature coefficients of resistance here

Nichrome - from Nichrome - Wikipedia

"Almost any conductive wire can be used for heating, but most metals conduct electricity with great efficiency, requiring them to be formed into very thin and delicate wires to create enough resistance to generate heat.

When heated in air, most metals then oxidize quickly, become brittle and break. Nichrome wire, when heated to red-hot temperatures, develops an outer layer of chromium oxide, which is thermodynamically stable in air, is mostly impervious to oxygen, and protects the heating element from further oxidation.

Nichrome alloys are known for their high mechanical strength and their high creep strength. The properties of nichrome vary depending on its alloy.

Uses: Because of its low cost of manufacture, strength, ductility, resistance to oxidation, stability at high temperatures, and resistance to the flow of electrons, nichrome is widely used in electric heating elements ... "

Nichrome wire, any wire, heats up because you are dissipating power in it.

Why use a high resistance wire?

'Most' power sources, at least ones that civilian hobbyists and experimenters have access to, are fairly limited current, and outputting at least several volts, like 1.5 or 2 V batteries, or a few amps from a bench power supply.

You must use a wire that's thick enough to support itself, say 0.5 mm, which is about 0.2 mm2. Now you have a choice of material - copper or nichrome, and maybe a few others like steel, brass, or some of the other 'resistance wires' like constantan.

If you do the sums for resistance for copper and nichrome, you get 85 mΩ/m for copper, 5.6 Ω/m for nichrome.

Now, which suits the available power sources you have? What can you power your 200 mm of foam cutter wire with?

If we disregard the choice of power supply, say we can choose the wire geometry freely, or order whatever supply we need, other considerations like the ability of the wire to withstand high temperatures come into play. The two most important for simple heating being the melting point, and the resistance to oxidation at high temperature. Nichrome outperforms both copper and steel in this respect, regardless of its resistivity.

If we want to use this as a resistor is a precision application, then the temperature coefficient of resistance becomes important. All elemental metal conductors have a similar and quite high tempco of resistance, 0.4% per C for copper. That is why most of the 'resistance' alloys like nichrome and constantan were developed. The clue is in the trade name 'constantan', an alloy that has constant resistance as you heat it.

The absence of $$\V\$$ in the equation $$\P=I^2R\$$ gives the impression that power is independent of voltage. This is false. The same goes for $$\P=\frac{V^2}{R}\$$, where it seems that current is not considered. Both of these equations are rooted in Ohm's law, which says:

$$V = I \times R$$

if you keep any one value fixed, and vary the second, then the third must also change in order to maintain the equality.

In the equation $$\P=I^2R\$$, as current $$\I\$$ changes, or resistance $$\R\$$ changes, you have failed to consider that there is also an implied change in voltage $$\V\$$.

The same goes for $$\P=\frac{V^2}{R}\$$; changing either $$\V\$$ or $$\R\$$ must result in a change of current $$\I\$$, which is implied, even though it's not not explicitly referred to here in the equation.

It may seem that an increase of resistance causes power to increase in one, yet decrease in the other, but that's because neither equation accounts for the third variable that changes sneakily, behind the scenes, according to Ohm's law.

Let's check some general physics law which says A=B/C. We really can say "the smaller C is, the bigger is A". Still in practical applications one can prefer to have quite big C although he still wants certain A. That can look especially obscure if bigger C costs more than small. This applies also to your case.

The glue can be found by checking what effort is needed to get the right B for the wanted A with the low cost small C. Or there's even more complex relations which should be taken into the account; for ex. manufacturability and durability.

In your case the cost of having right B for the wanted A with small C is one certain practical problem. P=(U^2)/R forces to use lower voltage for certain P if one makes R smaller. The voltage is often already given, for ex. there's 110V mains AC available. Reducing it causes losses and needs some equipment.

One can say "Then let's use U=110VAC". I may have more power than I need, but what's then? It at least is enough. The consequences can be fatal (fire) and the extra power anyway must be paid.

For a fixed voltage - you do need lesser and lesser resistance to generate more heat, as P = I^2R, and the I^2 goes up faster than the R goes down.

Thus for fixed V of 100V, 100 ohm nets 100 watts, and 10 ohm nets 1000 watts

The assumption is that the source can supply the power.

For a real power source with internal resistance Rint, the max power drawn occurs when Rload = Rint, this is not the most efficient way to draw power from a source however.

• Welcome to EE.SE. The question / answer editor supports html entities and some HTML. Use <sup>2</sup> (superscript tag) for 'squared'. It also supports MathJAX. Apr 30, 2023 at 21:43

The melting point of the material used in the heating element, as well as cost are factors considered here. Your thinking is correct, but what you did not consider is that the power supplies used in these types of devices are limited in their output current. While copper would generate heat quicker compared to a similar sized nichrome element, the output current would likely exceed the power supplies' safety limit

So P = IV

and V = IR

Then:

P = IV, substitute for V with V = IR then:

P = I (IR) = I^2R

And do the same substituting I = V/R...

• @timur then perhaps you are not sure of what you need to ask... will those relationships change with copper wire? Aluminium? Apr 30, 2023 at 9:41
• @timur the equations explained : With fixed U, a smaller R causes more current I, which causes more power P to nichrome and it heats up more. Is this what you are after? The resistance just needs to be suitable, for the voltage U and current I the supply can provide, for the heat you want to generate. There is no paradox. Apr 30, 2023 at 10:08
• @timur Copper wires have low R, nichrome has high R. For the same current I, voltage drops very little in copper wiring, and almost all voltage drops in nichrome wire. The copper wires are good in delivering power to load and bad heater, it would require huge current to heat low R wiring. The nichrome is just a resistive load which heats up most when least power is lost in copper wires delivering power to load. Apr 30, 2023 at 10:21
• @Justme: ok so nichrome wire is used primarily to drop almost all voltage on itself? this strategy may also backfire, since too much resistance would reduce the current right? Apr 30, 2023 at 10:26
• @timur Correct. You use a suitable length or nichrome to get a suitable R, to allow your U volt power supply to provide enough I and thus P for your purposes, and not overload the supply with too much current. Apr 30, 2023 at 10:29

You decide how much heat you want to dissipate, it gives you the resistance. For example 10 ohms

• You can use a very long copper cable : Expensive, heavy.

• You can use a thin copper cable : It will heat so much that it will melt. Very fragile.

• You use some resistive wire, for example nichrome or kanthal, which has just the right resistance for a reasonably sized heating element at low cost.