Video Transcript
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 𝜃 is the full-scale
deflection of the galvanometer. Two resistors, 𝑅 one and 𝑅 two,
are connected to the ohmmeter so that their resistances can be measured by the
ohmmeter. The galvanometer’s angle of
deflection is reduced by the angle 𝜙 when 𝑅 one is connected and its angle is
reduced by 𝛼 when 𝑅 two is connected. Which of the following correctly
relates the resistances of 𝑅 one and 𝑅 two? (A) 𝑅 one equals 𝑅 two, (B) 𝑅
one is less than 𝑅 two, or (C) 𝑅 one is greater than 𝑅 two.
So in this question, we’ve been
given a circuit diagram of an ohmmeter. And we’re also shown the same
circuit diagram, but this time with a resistor 𝑅 one connected in series and then
the same circuit again, but this time with a resistor 𝑅 two in place of 𝑅 one. So let’s start just by reminding
ourselves that an ohmmeter is a device which measures the resistance of a component
such as 𝑅 one or 𝑅 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 𝐺 in a
circle, indicates the value of the resistance.
Now, at this point, it’s 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 we’ll end up with a larger current in the circuit.
This relationship is summed up by
Ohm’s law, 𝐼 equals 𝑉 over 𝑅. If we consider 𝐼 to be the current
in the circuit, 𝑉 to be the voltage that we’re applying to the circuit, and 𝑅 to
be the total resistance of the circuit, then we can see that by increasing 𝑅, the
resistance, by a certain amount, we’ll decrease 𝐼, 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 𝜃.
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 it’s 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 galvanometer’s 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, we’re told that
connecting a resistor 𝑅 one in series with the ohmmeter will cause the needle’s
deflection to decrease by an angle of 𝜙. And we’re also told that connecting
a resistor 𝑅 two to the ohmmeter will cause the needle’s deflection to decrease by
an angle of 𝛼. Crucially, we’ve been told that 𝛼
is greater than 𝜙. In other words, connecting 𝑅 two
to the ohmmeter causes the needle’s deflection to decrease by a greater amount. This means that the resistor 𝑅 two
must be decreasing the total current in the circuit by a bigger amount than 𝑅 one
does. Therefore, we can conclude that 𝑅
two is greater than 𝑅 one or equivalently 𝑅 one is less than 𝑅 two.
If a galvanometer’s angle of
deflection is reduced by an angle 𝜙 when 𝑅 one is connected and its angle is
reduced by 𝛼 when 𝑅 two is connected and 𝛼 is greater than 𝜙, then we can
conclude that the resistance of 𝑅 one is less than the resistance of 𝑅 two.
