# Ohm's law holds at constant temperature - what about Joule heating?

Ohm's law states that the resistance of a conductor is constant provided its physical conditions, such as temperature, remain constant. But what I'm thinking is that as you increase the voltage across the conductor, it will heat up according to $I^2R$ (Joule heating), and so the temperature of it is increasing, so technically even if it turns out to display $I\propto V$, we can't say it obeys Ohm's law because Ohm's law needs its temperature to remain constant!

Where have I gone wrong in my thinking?

• If the change in resistance is solely due to temperature it still obeys the ohm law. But there are materials whose resistance will change with voltage even in fixed temperature. – Azad May 16 '15 at 18:02
• You haven't gone wrong. An incandescent light bulb, for example, does not obey Ohm's Law. There have even been some clever circuit designs that take advantage of that fact. The electronic components known as "resistors" do not experience the same extremes of temperature as does the filament of an incandescent bulb, and they come close enough to pure Ohm's Law behavior to satisfy most circuit designs. – Solomon Slow May 31 '16 at 14:26

The reason for restricting temperature change is that some materials exhibit a change in resistity when the temperature changes. If the resistivity is constant versus temperature the resistance won't change. In that case, there is no need to restrict the temperature.

A resistor is ohmic if it exhibits a constant slope V vs I curve. That resistor obeys Ohm's Law.

A light bulb filament won't obey Ohm's Law for a set of different DC voltages. But, for a moderate frequency (60 Hz) AC voltage, it will behave ohmically because the temperature, and hence, resistance, will stabilize at an equilibrium value. If the frequency drops to 1 Hz, the V vs I curve exhibits a lot of hysteresis and the V vs I slope can actually be negative due to the temperature fluctuations in the wire as it heats and cools in response to the slowly changing current.

You are correct that the resistance in some materials will increase as the temperature increases, and that the temperature can increase as a result of current flow. It still obeys Ohm's Law, however, as the relationship that Ohm's Law describes still holds. In other words, as the resistance increases, the current will decrease, just as Ohm's Law says it will.

If you think about that a little, you will realize that the current (and temperature) will stabilize under the right conditions, because an increase in temperature will decrease the current, which will decrease the temperature.

Technically, you can maintain a constant (or near-constant) temperature of the conductor in spite of Joule heating by cooling it with temperature regulation. Practically, resistance typically does not change dramatically under moderate Joule heating.

Resistors are usually colour coded and typically have three or four bands some times more). There are three relevant points here. The first is that the colour code of the first three bands gives the value of the resistance. The fourth band gives the tolerance (1%, 5%, 10%), which gives the range of possible values of the resistor. Resistors are mass produced components and a manufacturer isn't willing to claim that the resistance is exactly, say, $2.5645698.... \Omega$. rather they give a mean value (the colour coded value) and the tolerance, which is the range over what the value of the resistor actually is so that for a $10\%$ resistor and the actual resistance is (likely to be) some particular value between about $(2.25 - 2.75) \Omega$.

When you design a circuit it would be pointless if it would only work if its values was $2.5645698.... \Omega$. So, any change in resistance due to heating has no effect on the circuit. For high precision resistors there is sometimes a fifth band that gives the coefficient of variation with resistance with temperature (presumably if that is needed to take account of). As well, discrete component resistors are quite bulky components and can have a relatively large surface to dissipate heat.

As mentioned elsewhere, there are special types of resistors called thermistors, to see how they are used look here.

Bro the thing is Ohm's law is correct since $v$ directly proportional to $I$.

According to Ohm's law, temperature must remain constant, so$$\left[\text{heat}\right]~=~v {\times} I {\times} t \,,$$where time, $t$, is constant. Therefore, heat is directly proportional to $v×I$, and you increase $v$ then $I$ will also increase therefore since $v$ and $I$ have increased therefore heat (temperature) will also increase.

For example,$$2 {\times} \left[\text{heat}\right] ~=~2 {\times} v {\times} 2 {\times} I ~=~2 \left(v {\times} I \right) \,,$$therefore$$\frac{2 {\times} \left[\text{heat}\right]}{2}~=~v {\times} I \,,$$so$$\left[\text{heat}\right]~=~v {\times} I \,,$$therefore temperature remained constant, and therefore Ohm's law is obeyed.

But there are always exceptions.

when temperature vary the current is inversely proportional to the voltage thats why we use high voltage for transmission of power so as to over come copper loss i.e. i2r instead of current

• What you have mentioned is a special case for power transmission. The OP is asking for the general validity of Ohm's law. – Bruce Lee Aug 20 '16 at 13:44