From the circuit below, if the
ammeter reading is three amperes, calculate 𝑅. (A) Nine ohms, (B) three ohms, (C)
four ohms, (D) two ohms.
In the circuit diagram, we are
given a cell which is providing a potential difference of 18 volts across the
circuit. The ammeter is connected to measure
the current in the circuit. It gives a reading of three
amperes, which means that there is a current of three amperes through the part of
the circuit containing the ammeter. However, since there are two
resistors connected in parallel in this circuit, the current through each of the
parallel paths will have a different value than three amperes.
So, to calculate the resistance 𝑅,
we can replace the two resistors in parallel with a single equivalent resistor in
series with a resistance of 𝑅 total. Making this replacement, our
equivalent circuit looks like this. Recall that for a number of
components along parallel paths, the total resistance 𝑅 total is given by the
formula one over 𝑅 total equals one over 𝑅 one plus one over 𝑅 two plus one over
𝑅 three, and so on. Since we have two resistors in
parallel, this simplifies to the equation one over 𝑅 total equals one over 𝑅 one
plus one over 𝑅 two.
Now, we can substitute in the two
values for our resistors and rearrange to get 𝑅 total. Resistor one has a resistance 𝑅
one equal to 𝑅, and resistor two has a resistance 𝑅 two equal to two 𝑅. So, this means that one over 𝑅
total equals one over 𝑅 plus one over two 𝑅. We can multiply the one over 𝑅
term with two divided by two so that both terms on the right-hand side have the same
denominator and can be added together. Adding together these terms, two
over two 𝑅 plus one over two 𝑅 equals three over two 𝑅. We have then that one over 𝑅 total
is equal to three over two 𝑅.
We can now take the reciprocal of
both sides of this equation to obtain 𝑅 total. This gives us 𝑅 total equals two
𝑅 divided by three. This means that we have now
replaced two resistors with resistances 𝑅 and two 𝑅 in parallel by a single
resistor with a resistance of two 𝑅 over three. This single resistor is connected
in series with the cell and the ammeter.
We can now use Ohm’s law for the
single resistor in order to find the value of 𝑅. Recall that Ohm’s law can be
written as 𝑉 is equal to 𝐼 times 𝑅, where 𝑉 is the potential difference, 𝐼 is
the current, and 𝑅 is the resistance. Substituting in our values, we have
18 volts equals three amperes multiplied by two 𝑅 over three. Dividing both sides by three
amperes gives two 𝑅 divided by three is equal to 18 volts divided by three
amperes. 18 volts over three amperes works
out as six ohms. So, we have two 𝑅 over three
equals six ohms. Finally, multiplying both sides by
three over two, we have that 𝑅 equals three over two multiplied by six ohms, which
comes out as nine ohms.
Looking at the four possible answer
options, we see that our result matches the value in option (A). This means that option (A) is the
correct answer. The resistance 𝑅 is equal to nine