Video Transcript
Consider the circuit shown
below. What is the reading on the
voltmeter? (A) 12 volts, (B) six volts, (C) 18
volts, (D) nine volts.
To find the reading on the
voltmeter, we need to work out the potential difference across this resistor. We will begin by labeling each of
the resistors as follows. The current through the resistor 𝑅
one is equal to the total current in the circuit. If we can find this current, we can
use Ohm’s law to calculate the potential difference across resistor 𝑅 one. This potential difference will then
be equal to the reading on the voltmeter.
First, we must find the equivalent
resistance of the circuit. We can see that resistors 𝑅 two
and 𝑅 three are connected in parallel. Recall that for any number of
resistors in parallel, the total resistance of the parallel combination is given by
𝑅 total is equal to one over 𝑅 one plus one over 𝑅 two plus one over 𝑅 three et
cetera all to the power of negative one.
We can replace resistors 𝑅 two and
𝑅 three with an equivalent resistor, which we’ll call 𝑅 𝐴. This resistance is given by 𝑅 𝐴
is equal to one over 𝑅 two plus one over 𝑅 three all to the power of negative
one. By substituting in the values of 𝑅
two is equal to 𝑅 and 𝑅 three is equal to 𝑅, we find that 𝑅 𝐴 is equal to one
over 𝑅 plus one over 𝑅 all to the negative one. One over 𝑅 plus one over 𝑅 equals
two over 𝑅. So we have that 𝑅 𝐴 equals two
over 𝑅 to the negative one, which works out as 𝑅 over two.
We now have two resistors, 𝑅 𝐴
and 𝑅 one, that are connected in series. Recall that for any number of
resistors in series, the total resistance 𝑅 total is equal to the sum 𝑅 one plus
𝑅 two and so on for all the series resistances. So, we can replace resistors 𝑅 𝐴
and 𝑅 one by an equivalent resistor 𝑅 tot with resistance given by 𝑅 tot is equal
to 𝑅 𝐴 plus 𝑅 one. By substituting in the values of 𝑅
𝐴 equals 𝑅 over two and 𝑅 one equals 𝑅, we find that 𝑅 tot equals 𝑅 over two
plus 𝑅, which works out as three 𝑅 over two.
We can now use Ohm’s law across the
single resistor 𝑅 tot to find the value of current in this circuit. Recall that Ohm’s law can be
written as 𝑉 equals 𝐼 times 𝑅, where 𝑉 is the potential difference, 𝐼 is the
current, and 𝑅 is the resistance. Here, we want to calculate the
current through the circuit, so we need to rearrange Ohm’s law to make 𝐼 the
subject. To do this, we simply divide both
sides of the equation by the resistance 𝑅. This gives us that 𝐼 is equal to
𝑉 divided by 𝑅. We are given a cell 𝑉 𝐵, which
provides a potential difference of 18 volts across the circuit.
By substituting in the values of
the potential difference 𝑉 𝐵 equals 18 volts and the total resistance 𝑅 tot
equals three 𝑅 over two, we can use Ohm’s law to find the current through the
single resistor 𝑅 tot. This current 𝐼 is equal to 𝑉 𝐵
over 𝑅 tot, which is 18 volts divided by three 𝑅 over two ohms. This is equal to 36 divided by
three 𝑅 with units of amperes. This then works out as 12 over 𝑅
amperes. This current we have calculated
through the single equivalent resistor 𝑅 tot is equal to the total current through
the circuit.
Let’s now return to our original
circuit diagram. The total current that we just
calculated is equal to the current in resistor 𝑅 one. We can now use Ohm’s law to find
the value of the potential difference across the single resistor 𝑅 one. This potential difference will be
equal to the reading on the voltmeter. We are given the resistance of
resistor 𝑅 one as 𝑅, and we calculated a value of the current 𝐼 equal to 12
divided by 𝑅. By substituting these values into
Ohm’s law, we find that the potential difference across the resistor 𝑅 one is given
by 𝑉 equals 𝐼 times 𝑅, which is 12 over 𝑅 amperes multiplied by 𝑅 ohms. This works out as 12 volts.
We’ve already noted that the
voltmeter measures the potential difference across the resistor 𝑅 one. So this means that the reading on
the voltmeter will be equal to 12 volts. This corresponds to the value given
in option (A). So the correct answer is given in
option (A). The reading on the voltmeter is 12
volts.
