Video Transcript
A galvanometer has a coil of
resistance 25 ohms. What is the value of the resistance
of the multiplier resistor that is required to allow the galvanometer to measure
potential difference that is five times the potential difference across its
coil? (A) Five ohms, (B) 20 ohms, (C) 100
ohms, (D) 125 ohms.
The question wants us to calculate
the value of the multiplier resistance combined with a galvanometer used to convert
it into a voltmeter. Let’s recall that a multiplier
resistor must be connected in series with the galvanometer in order to convert the
galvanometer into a voltmeter. The question tells us the
resistance of the galvanometer. Labeling this as 𝑅 sub G, we have
that 𝑅 sub G is equal to 25 ohms.
We are also told the maximum value
of the potential difference that the voltmeter to be constructed must be able to
measure. This maximum measurable potential
difference, which we’ll call 𝑉 sub max, is five times the potential difference
across the galvanometer coil, which we’ll call 𝑉 sub G.
We want to find the value of the
multiplier resistance 𝑅 sub M. To do this, we can start by
recalling that there’s an equation for the maximum value of potential difference the
voltmeter can measure, which comes from Ohm’s law. This equation says that 𝑉 sub max
is equal to the current 𝐼 sub G through the galvanometer multiplied by 𝑅 sub M
plus 𝑅 sub G. On the left-hand side of the
equation, we know that 𝑉 sub max is equal to five times 𝑉 sub G, where 𝑉 sub G is
the potential difference across just the galvanometer coil.
Using Ohm’s law again, 𝑉 sub G can
be written as 𝑉 sub G equals 𝐼 sub G multiplied by 𝑅 sub G. So then 𝑉 sub max, which is equal
to five times 𝑉 sub G, must be equal to five times 𝐼 sub G times 𝑅 sub G. We then have that 𝑉 sub max is
both equal to five times 𝐼 sub G times 𝑅 sub G and to 𝐼 sub G multiplied by 𝑅
sub M plus 𝑅 sub G. Therefore, it must be true that
five times 𝐼 sub G times 𝑅 sub G is equal to 𝐼 sub G times 𝑅 sub M plus 𝑅 sub
G.
Expanding out the bracket on the
right and then subtracting 𝐼 sub G times 𝑅 sub G from both sides, we have that
four times 𝐼 sub G times 𝑅 sub G is equal to 𝐼 sub G times 𝑅 sub M. From here, we see that the factor
𝐼 sub G appears on both sides. So we can cancel this out. We have then that 𝑅 sub M is equal
to four times 𝑅 sub G. We know that 𝑅 sub G is equal to
25 ohms.
Substituting this into the equation
for 𝑅 sub M, we have that 𝑅 sub M is equal to four times 25 ohms. This works out as 100 ohms. We can see that this value matches
the one given in option (C). The correct answer is therefore
option (C). The resistance of the multiplier
resistor must be 100 ohms.
