### Video Transcript

In this video, weโre going to be
looking at the design of the ohmmeter.

An ohmmeter is a device which is
used to measure the electrical resistance of a component. In circuit diagrams, we can
represent an ohmmeter with an uppercase letter ฮฉ in a circle. And in this circuit diagram, the
ohmmeter is being used to measure the resistance, ๐
๐ฅ, of a test resistor. In this video, weโll see how we can
construct an ohmmeter using a cell, a galvanometer, a variable resistor, and a fixed
resistor connected in series.

So to start things off, letโs
consider a resistor. Letโs say that this resistor has
some resistance, ๐
๐ฅ, and we want to build an ohmmeter so we can measure the value
of ๐
๐ฅ. Well, Ohmโs law tells us that the
resistance of a component is given by the voltage across that component divided by
the current in that component. So as a general idea, if we could
apply some known voltage ๐ to a resistor and then measure the current ๐ผ, then we
could calculate its resistance by dividing the voltage by the current.

So with this in mind, letโs connect
a cell to our resistor. Now that weโre applying a potential
difference, letโs call this ๐, to our resistor, weโll find that thereโs a current
in the circuit, and we can call this current ๐ผ. If our voltage is known, then all
we need to do is measure the current ๐ผ and we can calculate the resistance using
Ohmโs law. To help us measure the current,
letโs introduce a galvanometer to our circuit. And letโs quickly recall that a
galvanometer is a device which can measure the magnitude and direction of a current
using a needle and a dial.

As a quick side note, because most
galvanometers measure current in both directions, the zero tends to be in the middle
of the scale. Since the ohmmeter weโre trying to
build is a direct current circuit, that is, a circuit where current only goes in one
direction, we can modify our galvanometer so that the zero is at one end of the
scale. So now things are set up so that
the needle will deflect this way in response to a current in this direction in our
circuit. With this slightly modified
galvanometer in place, we now have some way of getting information about the current
in the circuit and therefore the resistance of our test resistor.

Because weโre trying to build an
ohmmeter, we want to change the dial on our galvanometer so that it shows resistance
rather than current. Ohmโs law tells us that if the
current in our circuit is really small, then the resistance must be really big. And similarly, if weโre getting a
really big current, then it must mean that the resistance is small. Taking this idea to its extreme, we
could say that if a circuit has zero current, then it must have effectively infinite
resistance. This would be equivalent to putting
a break in our circuit, which would make it impossible for a current to exist.

So since zero current means that
resistance is infinite, then we could change the dial on our galvanometer so that
when the dial is in this position, instead of reading zero current, it reads
infinite resistance. So now if this end of our scale
corresponds to infinite resistance, that means, ideally, we want the other end of
the scale of our ohmmeter to correspond to zero resistance.

Now we know that galvanometers are
very sensitive instruments. So they generally reach maximum
deflection in this direction for a pretty small current, usually in the region of
microamps or milliamps. We can call this current ๐ผ ๐. If we want to set up our ohmmeter
so that the needle reaches this position when the test resistance is zero, that
means we need to make some modifications to our circuit so that when the value of
the test resistance is zero, the value of the current in the circuit is ๐ผ ๐, the
maximum deflection current of the galvanometer.

To achieve this, we need to wire in
a couple of resistors to our circuit, a variable resistor represented by a resistor
sign with a diagonal arrow through it and an ordinary fixed resistor. And we can say that the resistance
of the variable resistor is ๐
๐ and the resistance of the fixed resistor is ๐
๐น. While weโre on the topic of
resistors, itโs important to remember that the galvanometer has its own internal
resistance, ๐
๐บ. The function of these additional
resistors is to ensure that when the value of this test resistance is zero, the
current in the galvanometer is just enough to cause maximum deflection of the
needle.

We can now do some calculations to
determine what these values of resistance need to be in order to make this the
case. Since weโre now considering the
situation where the test resistance is zero, this is equivalent to replacing the
test resistor with just a wire. So since all of this stuff is
effectively our ohmmeter, weโre effectively short-circuiting the ohmmeter. To work out the values of resistors
that we need to use, we can use Ohmโs law. We want to construct our ohmmeter,
so that the total effective resistance of this circuit when the test resistance is
equal to zero is just sufficient to limit the current to ๐ผ ๐. In other words, weโre looking to
make the total resistance of our ohmmeter, which we can call ๐
ฮฉ, satisfy this
equation.

Letโs also recall that when we have
several resistors connected in series, the total effective resistance, ๐
๐, is
equal to the sum of the individual resistances. This means that the effective total
resistance of all of our ohmmeter components connected in series is equal to ๐
๐,
the resistance of the variable resistor, plus ๐
๐น, the resistance of the fixed
resistor, plus ๐
๐บ, the resistance of the galvanometer. So we can replace ๐
ฮฉ in our
expression with ๐
๐ plus ๐
๐น plus ๐
๐บ. If we now subtract ๐
๐น and ๐
๐บ
from both sides of this equation, weโre left with this expression, which enables us
to calculate the value that we need to set our variable resistor to in order to
correctly calibrate the position of the zero on our ohmmeter.

Practically speaking though because
a variable resistor is by its nature easily adjusted, we could correctly calibrate
our variable resistor simply by decreasing it from its maximum resistance until the
needle on the ohmmeter reaches the maximum deflection. And at this point, weโll know that
the maximum deflection on the dial corresponds to a test resistance value of
zero. So we can confidently write a zero
at this end of the scale. So we now have a completely
assembled and correctly calibrated ohmmeter. If we reintroduce a test resistor,
then the needle on the dial will change to indicate its resistance. However, here we run into a
problem. Our ohmmeter measures values of
resistance from โ to zero. However, we donโt know what any of
these values in the middle of the dial actually correspond to.

Luckily, we can figure out the
scale on our ohmmeter by using the fact that the deflection of the needle on a
galvanometer is proportional to the current. This means that if a current ๐ผ ๐
is sufficient to cause full deflection of a needle, then a current half this size
will cause half deflection of the needle, putting it in exactly the middle of the
dial. Similarly, a current of a quarter
๐ผ ๐ would cause the needle to deflect a quarter of the way round and so on.

Now, if we rearrange Ohmโs law to
make ๐ผ the subject, we obtain the expression ๐ผ equals ๐ over ๐
. Since ๐ in our circuit is a
constant, this means that ๐ผ, the current in our circuit, is inversely proportional
to ๐
๐, the total resistance of our circuit. This means, for example, if we
multiply the total resistance by two, then the current will halve or if we were to
multiply the total resistance by four, then we would divide the total current by
four.

Now, weโve already shown that to go
from full deflection of the needle which occurs when the test resistance is zero to
a half deflection of the needle, we would need to halve the current. And Ohmโs law tells us that in
order to halve the current, we would need to double the total resistance of the
circuit. This means that if we add a test
resistor and the needle moves to half deflection, then adding this test resistor
must have doubled the resistance of the entire circuit.

This would mean that the resistance
of the test resistor is exactly equal to the resistance of the ohmmeter, which means
the resistance measured by the halfway position on the scale is equal to ๐
ฮฉ, the
resistance of the ohmmeter itself which, as weโve shown, is equal to the resistance
of the variable resistor plus the resistance of the fixed resistor plus the
resistance of the galvanometer.

Following the same reasoning, the
resistance indicated by this position on the dial would be half the resistance of
the ohmmeter. And the resistance indicated by
this position on the dial would be twice the resistance of the ohmmeter. So we can see that because the
deflection of the galvanometerโs needle is proportional to current but current is
inversely proportional to resistance that the scale on our ohmmeter is
nonlinear. That is, the deflection is not
proportional to the resistance that weโre measuring.

Okay, now that weโve seen how to
assemble and calibrate an ohmmeter and how to interpret the reading on the dial,
letโs have a go at answering a question.

A circuit that can be used as an
ohmmeter is shown. The circuit uses a galvanometer, a
direct current source with a known voltage, a fixed resistor, and a variable
resistor. The angle ๐ is the full-scale
deflection of the galvanometer. Two resistors, ๐
one and ๐
two,
are connected to the ohmmeter so that their resistances can be measured by the
ohmmeter. The galvanometerโs angle of
deflection is reduced by the angle ๐ when ๐
one is connected and its angle is
reduced by ๐ผ when ๐
two is connected. Which of the following correctly
relates the resistances of ๐
one and ๐
two? (A) ๐
one equals ๐
two, (B) ๐
one is less than ๐
two, or (C) ๐
one is greater than ๐
two.

So in this question, weโve been
given a circuit diagram of an ohmmeter. And weโre also shown the same
circuit diagram, but this time with a resistor ๐
one connected in series and then
the same circuit again, but this time with a resistor ๐
two in place of ๐
one. So letโs start just by reminding
ourselves that an ohmmeter is a device which measures the resistance of a component
such as ๐
one or ๐
two. In order to measure the resistance
of a component, we connect it in series with the ohmmeter. The deflection of the needle on a
galvanometer, which is represented in our circuit diagrams as a capital ๐บ in a
circle, indicates the value of the resistance.

Now, at this point, itโs useful to
remember that the needle in a galvanometer actually responds to current. The idea behind an ohmmeter is that
by applying a known voltage to a circuit containing a test resistor and a
galvanometer, the needle on the galvanometer will respond to the amount of current
in the circuit. We then know that if the test
resistor has a really large resistance, then only a small current will exist in the
circuit. Conversely, if the resistor has a
very low resistance, then weโll end up with a larger current in the circuit.

This relationship is summed up by
Ohmโs law, ๐ผ equals ๐ over ๐
. If we consider ๐ผ to be the current
in the circuit, ๐ to be the voltage that weโre applying to the circuit, and ๐
to
be the total resistance of the circuit, then we can see that by increasing ๐
, the
resistance, by a certain amount, weโll decrease ๐ผ, the current, by a proportional
amount. In other words, the current in the
circuit and the total resistance of the circuit are inversely proportional to one
another. Now, if we look at the diagram on
the left, we can see that the needle on the galvanometer is deflected fully. And incidentally, the angle of this
deflection has been called ๐.

Now, a given galvanometer will have
some current which causes maximum deflection of the needle. And we generally find this is in
the milliamp or microamp range. Any current smaller than this will
only cause a partial deflection of the needle, enabling the galvanometer to
effectively measure that current. But any current greater than the
full deflection current will just cause the needle to be fully deflected. In other words, a galvanometer on
its own is only useful for measuring current within a small given range. And this is where these resistors
come into play.

The function of the variable and
the fixed resistors are to ensure that the ohmmeter on its own has just enough
resistance such that the current is just big enough to cause maximum deflection of
the needle. And once this is achieved, we can
say that the ohmmeter has been calibrated. Once this has been done, then
adding a resistor in series to the ohmmeter will increase the total resistance of
the circuit and therefore decrease the current such that itโs now less than the
current which would cause full-scale deflection of the galvanometer.

And at this point, it might be
useful to remind ourselves that when we connect resistors in series, the total
resistance is simply the sum of the resistances of the individual resistors in the
circuit. So we know that connecting a
resistor in series with the ohmmeter increases the overall resistance of the circuit
and therefore causes the galvanometerโs needle to back away from full deflection due
to a drop in current. The bigger the value of the
resistor that we connect in series with the ohmmeter, the more the deflection of the
needle of the galvanometer will decrease by.

In this question, weโre told that
connecting a resistor ๐
one in series with the ohmmeter will cause the needleโs
deflection to decrease by an angle of ๐. And weโre also told that connecting
a resistor ๐
two to the ohmmeter will cause the needleโs deflection to decrease by
an angle of ๐ผ. Crucially, weโve been told that ๐ผ
is greater than ๐. In other words, connecting ๐
two
to the ohmmeter causes the needleโs deflection to decrease by a greater amount. This means that the resistor ๐
two
must be decreasing the total current in the circuit by a bigger amount than ๐
one
does. Therefore, we can conclude that ๐
two is greater than ๐
one or equivalently ๐
one is less than ๐
two.

If a galvanometerโs angle of
deflection is reduced by an angle ๐ when ๐
one is connected and its angle is
reduced by ๐ผ when ๐
two is connected and ๐ผ is greater than ๐, then we can
conclude that the resistance of ๐
one is less than the resistance of ๐
two.

Letโs now review the key points
that weโve learned in this video. Firstly, weโve seen that an
ohmmeter can be made by connecting a direct current source, a fixed resistor, a
variable resistor, and a galvanometer in series with each other. Weโve also showed that to calibrate
the resistor, the resistances of the fixed and variable resistors must be chosen
such that when the ohmmeter is short-circuited, the current is equal to the
full-scale deflection current of the galvanometer. And finally, weโve seen that
ohmmeters have a nonlinear scale, which varies from infinite resistance indicated by
zero deflection of the galvanometerโs needle to zero resistance, which is indicated
by full deflection. This is a summary of the design of
the ohmmeter.