Video Transcript
In the circuit shown, 𝑅 two equals
two 𝑅 one and 𝑅 three equals three 𝑅 two. The total current through the
circuit is 0.36 amperes. What is the resistance 𝑅 one?
In the circuit shown, we have three
resistors in parallel and we want to find the value of the resistance, 𝑅 one. To find the resistance 𝑅 one, we
can convert the three resistors in parallel into a single equivalent resistor 𝑅
total. This will allow us to find the
total resistance of the circuit in terms of 𝑅 one. Since we are given both the
potential difference provided by the cell and the total current through the circuit,
we can also use Ohm’s law to calculate the total resistance in the circuit. Then, we can use this value to
deduce 𝑅 one.
Let’s start by finding the
equivalent resistance of the circuit, 𝑅 total. Recall that for any number of
resistors in parallel, the total resistance is given by 𝑅 total equals one over 𝑅
one plus one over 𝑅 two et cetera plus one over 𝑅 𝑁 to the negative one
power. So, for this circuit, we can
replace the three resistors that are connected in parallel with a single equivalent
resistor which will have a resistance 𝑅 total, which is equal to one over 𝑅 one
plus one over 𝑅 two plus one over 𝑅 three to the negative one power.
We are given that 𝑅 two equals two
𝑅 one, and we are also given that 𝑅 three equals three 𝑅 two. Because we are trying to find the
value of 𝑅 one, it’s useful to find 𝑅 three in terms of 𝑅 one. To do this, we can substitute the
value of 𝑅 two into the expression for 𝑅 three to get 𝑅 three equals three times
two 𝑅 one, which is equal to six 𝑅 one. By substituting in the values for
𝑅 two and 𝑅 three, we can find an expression for 𝑅 total in terms of 𝑅 one. We find that 𝑅 total, which is one
over 𝑅 one plus one over 𝑅 two plus one over 𝑅 three to the negative one power,
is equal to one over 𝑅 one plus one over two 𝑅 one plus one over six 𝑅 one to the
negative one power.
By manipulating the algebra a
little bit, we can simplify this expression. First, we want to make sure that
all three terms have the same denominator, six 𝑅 one. To do this, we multiply the top and
bottom of the first term by six to get six over six 𝑅 one. Then, we can multiply the top and
bottom of the second term by three to get three over six 𝑅 one. This leaves us with six 𝑅 one in
the denominator of all three terms. Then, we can combine these terms
into a single fraction by summing the numerators. Six plus three plus one divided by
six 𝑅 one is 10 divided by six 𝑅 one. To get this fraction into its
simplest form, we can cancel a factor of two from the top and bottom. This gives us five divided by three
𝑅 one. Finally, taking the reciprocal, we
find 𝑅 total is equal to three 𝑅 one divided by five.
So, we’ve found an expression for
the total resistance in the circuit in terms of 𝑅 one. We can now use Ohm’s law to find
the value of the total resistance, which will then allow us to work out the value of
𝑅 one.
Recall that Ohm’s law can be
written as 𝑉 equals 𝐼𝑅, where 𝑉 is the potential difference, 𝐼 is the current,
and 𝑅 is the resistance. Here, we want to use Ohm’s law to
calculate the value of the resistance 𝑅 total. So we need to rearrange this
equation to make 𝑅 the subject. To do this, we simply divide both
sides by 𝐼, which leaves us with the equation 𝑉 over 𝐼 is equal to 𝑅. We are given a cell, which is
providing a potential difference of 18 volts across the circuit. And we are also given that the
total current through the circuit is 0.36 amperes. These values are the same through
the resistor 𝑅 total. So by substituting in the values of
potential difference and current, we can use Ohm’s law to find that 𝑅 total is 18
volts divided by 0.36 amperes, which is equal to 50 ohms.
We now have two expressions for 𝑅
total. 𝑅 total equals 50 ohms, and 𝑅
total equals three 𝑅 one divided by five. Equating these two expressions, we
find that three 𝑅 one divided by five equals 50 ohms. Multiplying both sides by five and
dividing both sides by three, we find that 𝑅 one equals 50 ohms times five-thirds,
which is 250 divided by three ohms or 83.333 and so on ohms. Therefore, to the nearest integer,
the value of the resistance 𝑅 one is 83 ohms.
