Video Transcript
Find the total current in the
circuit shown. Give your answer to one decimal
place.
In this question, we have a circuit
that contains both series and parallel combinations of resistors. We want to calculate the total
current in the circuit. We will begin by finding the
equivalent resistance of the circuit.
First, we will redraw the circuit
so that we can see more clearly how all of the components are connected. If we compare this circuit to the
one shown in the question, we can see that they’re equivalent. The 3.5-ohm and 1.8-ohm resistors
are connected in series with each other, and the 1.2-ohm resistor is connected in
parallel to them both. The 6.5-ohm resistor is connected
in series to the parallel combination of the other three resistors. We have also labeled the resistors
𝑅 one to 𝑅 four to make our calculations easier to follow. To find the equivalent resistance
of the circuit, we simply need to work out the total resistance provided by this
combination of resistors.
The first step in calculating the
equivalent resistance of the circuit is to find the equivalent resistance of the
resistors 𝑅 one and 𝑅 two, which are connected in series. Recall that for any number of
resistors in series, the total resistance, 𝑅 total, equals 𝑅 one plus 𝑅 two plus
dot dot dot plus 𝑅 sub 𝑁. We can replace resistors 𝑅 one and
𝑅 two with an equivalent resistor 𝑅 sub 𝐴 with resistance 𝑅 sub 𝐴 equals 𝑅 one
plus 𝑅 two. By substituting in the values for
resistors 𝑅 one equals 3.5 ohms and 𝑅 two equals 1.8 ohms, we find that 𝑅 sub 𝐴
equals 𝑅 one plus 𝑅 two equals 3.5 ohms plus 1.8 ohms equals 5.3 ohms. We can now see two resistors 𝑅 sub
𝐴 and 𝑅 three that are connected in parallel.
Recall that for any number of
resistors in parallel, the total resistance, 𝑅 total, equals the inverse of the
quantity one over 𝑅 one plus one over 𝑅 two plus and so on plus one over 𝑅 sub
𝑁. So we can replace resistors 𝑅 sub
𝐴 and 𝑅 three with an equivalent resistor 𝑅 sub 𝐵 with resistance given by 𝑅
sub 𝐵 equals the inverse of the quantity one over 𝑅 sub 𝐴 plus one over 𝑅
three. By substituting in the values of 𝑅
sub 𝐴 equals 5.3 ohms and 𝑅 three equals 1.2 ohms, we find that 𝑅 sub 𝐵 equals
the inverse of the quantity one over 𝑅 sub 𝐴 plus one over 𝑅 three equals the
inverse of the quantity one over 5.3 ohms plus one over 1.2 ohms, which, to three
decimal places, is 0.978 ohms.
Finally, we have two resistors 𝑅
sub 𝐵 and 𝑅 four connected in series. We can replace these two resistors
with an equivalent resistor 𝑅 sub 𝐶 equals 𝑅 sub 𝐵 plus 𝑅 four. Substituting in the values of 𝑅
sub 𝐵 equals 0.978 ohms and 𝑅 four equals 6.5 ohms, we get 𝑅 sub 𝐶 equals 0.978
ohms plus 6.5 ohms equals 7.478 ohms. The resistor 𝑅 sub 𝐶 has an
equivalent resistance to the four resistors initially shown in the question. We can now use Ohm’s law across the
single resistor 𝑅 sub 𝐶 to find the value of the current.
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’re interested in finding
the current, so we need to rearrange this equation to make 𝐼 the subject. To do this, we just divide both
sides by 𝑅 to get 𝐼 equals 𝑉 over 𝑅. We are given a cell which is
providing a potential difference of 14 volts across the circuit. Substituting in this value and the
value of the resistance that we just calculated, we can use Ohm’s law to find that
the current across the single resistor 𝑅 sub 𝐶 is equal to 𝐼 equals 𝑉 divided by
𝑅 sub 𝐶 equals 14 volts divided by 7.478 ohms equals 1.87204 and so on
amperes.
The current is the same throughout
a series circuit, so the current we have calculated across the single resistor is
equal to the total current through the equivalent circuit. Because the equivalent circuit has
the same total resistance as the original circuit we were given, this current must
also equal the total current in the original circuit.
The final bit we need to complete
this question is to give the answer to one decimal place. We will take the numbers before the
decimal place and the two numbers after the decimal place to get 1.87 amperes. Then, we look at the second number
after the decimal place to see if we need to round down or round up. We see that the second number after
the decimal place is seven, so we can round this up, which gives the answer of 1.9
amperes. Hence, the total current through
the circuit is 1.9 amperes.
