Temperature Dependence and the Colour Code

Transcript

0:00Introduction

Hello everyone! One small puzzle before we start. Heat an iron wire, and its resistance goes up. Heat a carbon resistor, and its resistance goes down! So, can we join the two so that heating changes nothing at all? Interesting, right? We will build exactly that, and also learn the colour code on resistors. This topic comes up often in JEE problems. So, let's begin.

0:30Metals: hotter means more resistance

First, recall the resistivity from the last lesson: rho, the Greek letter like a curly p, equals m, over n times e squared times tau, where the small two is a power. Here m is the mass of the electron, and e is its charge; n is the number of free electrons per unit volume, and tau, the Greek letter, is the relaxation time, the mean time between collisions. With me? Now, heat the metal, and think about the collisions. What happens to tau, and then to rho? Yes, exactly! The collisions become more frequent, so tau falls. And tau sits in the bottom of the formula, so rho rises. Okay? Over moderate ranges, R T equals R nought times, open bracket, one plus alpha times delta T, close bracket. R T, with a small subscript T, is the resistance after heating, and R nought, with a small zero, the resistance before; delta T, with delta the Greek triangle, is the rise in temperature. Alpha, the Greek letter, is the temperature coefficient of resistance, greater than zero for metals. Right? For copper, alpha is about four times ten to the power minus three per kelvin. The wavy equals sign means approximately, and capital K to the power minus one means per kelvin. Clear? But remember the condition: this straight line rule holds over moderate ranges only. Over a wide range of temperature, the rise is not linear. Got it?

2:01Alloys and semiconductors

Now, some alloys behave differently. Manganin, constantan and nichrome have a high rho, but a very small alpha, so their resistance hardly changes as they warm up. Okay? That is why these alloys are used for standard resistors and for heaters. Got it? Now, semiconductors. When they are heated, tau falls for them too. So their resistance also rises, right? No, no, wait, that is the trap! For semiconductors, alpha is less than zero, so alpha is negative. Heat them, and the resistance goes down. With me? Look at the formula for rho again. In a semiconductor, n, the number of carriers, rises steeply with temperature. That rise outweighs the fall in tau, so rho falls. Clear?

2:50Finding α from two readings

Now, a result that saves time. Write R equals R nought times, open bracket, one plus alpha times T, close bracket, with T in degrees Celsius, the small circle and capital C on screen. So R nought is the resistance at zero degrees Celsius. Okay? We measure R one at T one, and R two at T two; the small ones and twos are subscripts. Divide, and R nought cancels: R two over R one equals one plus alpha times T two, over one plus alpha times T one. Now cross-multiply, and find alpha yourself. Cross-multiplying gives R two, plus alpha times R two times T one, equals R one, plus alpha times R one times T two. Right? Collect the alpha terms, and alpha equals R two minus R one, over R one times T two, minus R two times T one. Look at the bottom: it is crossed, R one with T two, and R two with T one. Got it?

3:45Reading the colour code

Now, a different skill. Resistors carry their value as coloured bands, each colour a digit: black is zero, brown one, red two, orange three, and yellow four. Okay? Then green is five, blue six, violet seven, grey eight, and white nine. Right? Use this line to remember the order: B B Roy of Great Britain has a Very Good Wife. The first letters follow the colours: B B for black and brown, Roy for red, orange and yellow, and so on, up to white. Clear? Now, reading a resistor: the first two bands are the two digits, and the third band is the power of ten that multiplies them. With me? And the fourth band is the tolerance: gold means five percent, silver ten percent, and no fourth band means twenty percent. Okay? So let us read one together. The bands are yellow, violet, red and gold. What is the resistance? Yellow is four and violet is seven, so the digits are forty seven. Red is two, so we multiply by ten to the power two, and that gives four thousand seven hundred ohm, with the capital omega as the ohm symbol. That is four point seven kilo ohm, small k for kilo, and gold adds plus or minus five percent. Careful, the third band is a power of ten, not a digit. Got it?

5:08Example 1: 10 ohm coil heated from 20 C to 70 C

Alright, example one, an easy one. We have a coil of resistance ten ohm at twenty degrees Celsius, with alpha four times ten to the power minus three per kelvin. Find its resistance at seventy degrees Celsius. Pause the video and try it first. Done? Okay, let's check. The new temperature is seventy, so delta T is seventy, right? Oh no, careful, that is the trap here! Delta T is the rise, seventy minus twenty, which is fifty degrees Celsius, the same as a rise of fifty kelvin. Okay? So R equals ten times, open bracket, one plus four times ten to the power minus three times fifty, close bracket. Right? Four times ten to the power minus three, times fifty, gives zero point two. So R is ten times one point two, which is twelve ohm. So always subtract the starting temperature. Got it?

6:06Example 2: carbon and iron in series, R independent of T

Now example two, a harder one, and it answers our puzzle. Take a carbon resistor, with alpha equal to minus five times ten to the power minus four per kelvin, in series with an iron wire, with alpha equal to five times ten to the power minus three per kelvin. Find the ratio of their resistances for the combination to be independent of temperature. Pause the video and try it first. Done? Okay, let's check. In series, resistances add. The subscript C means carbon and F e means iron, with each resistance taken at zero degrees Celsius. So the total R is R carbon times, one plus alpha carbon times T, plus R iron times, one plus alpha iron times T, okay? Open the brackets and group the terms in T: R carbon plus R iron, plus, open bracket, R carbon times alpha carbon, plus R iron times alpha iron, close bracket, all times T, as on screen. So tell me, when does this total stop depending on temperature? Yes, exactly! When the bracket in front of T is zero: R carbon times alpha carbon, plus R iron times alpha iron, equals zero. This works only because carbon's alpha is negative, so its fall cancels the iron's rise. Right? So R carbon over R iron equals minus alpha iron over alpha carbon: minus five times ten to the power minus three, over minus five times ten to the power minus four. The minus signs cancel, giving ten, so carbon to iron is ten to one; the colon means is to. Careful, not one to ten: carbon's alpha is ten times smaller in size, so it needs ten times the resistance. And that solves our puzzle, okay?

7:53Quick revision: R vs T, alloys, semiconductors, colour code

So, quick revision. Metals: R T equals R nought times, one plus alpha times delta T, with alpha positive; semiconductors have a negative alpha. Manganin, constantan and nichrome have a very small alpha. And the colour code: two digits, a power of ten, then the tolerance. Now try the practice problems on this page. All the best!