Resistance of a Wire by Ohm's Law

Transcript

0:00Introduction

Hello everyone, here is a small puzzle before we start. You do the Ohm's law experiment, and you plot voltage against current. But your best fit straight line misses the origin, and it cuts the voltage axis at zero point one volt. So is your whole experiment ruined? We will answer that in this video. This belongs to experimental skills in JEE, and questions on it do come up. So, let's begin.

0:26Aim: the resistance of a wire

Look at the circuit on the left of the figure. Our aim is the resistance of a wire, capital R, the small box at the top. The circle marked capital A is the ammeter, with a low resistance, and it goes in series with the wire, okay? The circle marked capital V is the voltmeter, with a high resistance, and it goes in parallel with the wire. Right? The positive terminals of both meters go towards the positive terminal of the cell, the pair of short lines on the far left. Clear? The box with the arrow at the bottom is the rheostat. It changes the current, and that gives us a set of readings. Got it?

1:05The graph: what its slope tells you

Each pair of readings gives the resistance. Capital R equals capital V over capital I, the voltage across the wire over the current through it. With me? Now look at the graph on the right, with capital V going up and capital I going across, along the two arrows. The dots are our readings, on a straight line through the origin, and its slope is capital R, okay? Now here is a trap. Suppose a student plots capital I on the y axis against capital V, and the slope is zero point two five. The unit on screen, with the power minus one, means ampere per volt. So is the resistance zero point two five ohm, the omega sign on screen? No, it is not, so be careful. The axes are swapped, so the slope is now one over capital R, right? And one over zero point two five gives four ohms.

2:00Why the current stays small

In the lab, the current is kept small, and it is passed only while we take a reading. So let me ask you, why do we bother? If you said heating, that is correct, because the wire must not heat up. At large currents its resistance rises, so the same increase of current needs a larger increase of voltage. That is why the graph of capital V against capital I bends towards the voltage axis. Got it?

2:27Zero error, and back to our puzzle

Before starting, we note two things for each meter: its least count and its zero error. Okay? Take this ammeter, which reads zero point zero two ampere when no current flows, so its zero error is positive. Now it reads zero point four two ampere. So let me ask you, what is the true current? We subtract the zero error, so do not add it by mistake. Zero point four two minus zero point zero two, equals zero point four zero ampere. Right? Now back to our puzzle. The best fit line misses the origin, and it cuts the voltage axis at zero point one volt. So what does this suggest, and is the slope still useful? Very good if you said zero error, because it suggests a zero error of about zero point one volt in the voltmeter. Clear? And the slope is unaffected, so it still gives capital R, okay? So your experiment is not ruined at all, and that settles our puzzle.

3:28Where exactly should the voltmeter go?

Now, any real voltmeter draws some current, and any real ammeter drops some voltage. So where we join the voltmeter does matter, right? Take a resistance of exactly ten ohms, an ideal ammeter, and a voltmeter of one thousand ohms directly across it. The ammeter reads the current in the parallel combination, so capital V over capital I is ten times one thousand, over one thousand and ten. That gives nine point nine zero ohms, one per cent too low, okay? Now join the voltmeter across the resistor and a zero point five ohm ammeter together. It also reads the drop in the ammeter, so capital V over capital I is ten plus zero point five, which gives ten point five ohms, five per cent too high. With me? So which arrangement suits a very small resistance, and which a very large one? For a very small resistance, join the voltmeter directly across the wire, because it draws a negligible share of the current. Okay? For a very large resistance, join it across the wire and the ammeter together, because the ammeter's drop is then negligible. Clear?

4:37From R to resistivity, and the errors

Now from the resistance we get the resistivity, rho, this Greek letter. Rho equals capital R times capital A, over capital L, where capital A is the area of cross section, not the ammeter, and capital L is the length. With diameter small d, that becomes pi times small d squared times capital R, over four times capital L, okay? Now let us turn to the errors. The little triangle is capital delta, and delta R over R is the relative error in the resistance. It equals the relative error in the voltage, plus the relative error in the current, the two fractions on screen. Right? For the resistivity we add two more terms: delta L over L, and two times delta d over d. Now why is that two sitting with the diameter? Yes, exactly, it is there because small d is squared in the formula, so the error in the diameter counts twice. Got it?

5:34Example 1: least counts of 500 mA ammeter and 3 V voltmeter

Now the examples, the easier one first. An ammeter of range zero to five hundred milliamperes, the m A on screen, has fifty divisions. And a voltmeter of range zero to three volts has thirty divisions. Find their least counts. Pause the video and try it first. Done? Okay, let's check. The least count is the value of one division, so we divide the range by the number of divisions. Five hundred milliamperes over fifty gives ten milliamperes, right? For the voltmeter, three volts over thirty gives zero point one volt. Keep these two ready, because the next example uses the same meters. With me?

6:19Example 2: R with its error from 2.0 V and 0.40 A readings

Now the harder one, with those same two meters. Here, a student reads two point zero volts and zero point four zero ampere. Find capital R with its error, taking the error of each reading as one least count. Pause the video and try it first. Done? Okay, let's check. Capital R is two point zero over zero point four zero, which gives five point zero ohms. Okay? Now for the error. Since we divide the voltage by the current, do the two relative errors subtract? No, they do not, so be careful, because they add. One least count on the voltmeter is zero point one volt, and on the ammeter ten milliamperes, which is zero point zero one ampere. Zero point one over two point zero gives five per cent, and zero point zero one over zero point four zero gives two point five per cent. Together that is seven point five per cent, right? So delta R is seven point five per cent of five point zero, which gives zero point three seven five ohm. Rounding the error to one figure, that is approximately zero point four ohm, the wavy equals sign. So capital R is five point zero, plus or minus zero point four ohms. If you got this on your own, very well done!

7:43Quick revision: V-I slope, zero error, errors in R and rho

So, quick revision. First, ammeter in series, voltmeter in parallel, and the slope of capital V against capital I gives the resistance. Second, keep the current small, pass it only while reading, and note the least count and zero error first. Third, the relative errors in the voltage and the current add, and for the resistivity add delta L over L, plus two times delta d over d. Now go and try the practice problems on this page, and all the best!