For the circuit below, what is the
value of the resistance 𝑅? (A) 2.4 ohms, (B) 20.8 ohms, (C)
33.6 ohms, (D) 5.2 ohms.
We want to find the resistance 𝑅
for the resistor in the circuit shown. Because there are two cells in this
circuit, we cannot calculate the equivalent resistance of the three resistors in the
circuit. Therefore, we will need to use
Kirchhoff’s laws to calculate the resistance 𝑅 in the circuit.
We begin by labeling each of the
components in the circuit. This circuit has three loops that
can be followed in either direction, which we can label on the circuit diagram. To find the resistance 𝑅 of
resistor 𝑅 three, we can look at either loop one or loop three. Loop three has an unknown potential
difference 𝑉 one. So we will analyze this circuit
using loop one instead.
Recall that Kirchhoff’s second law
states that the sum of the potential difference across each component in a loop is
equal to zero. We can label the potential
difference across resistor 𝑅 two as 𝑉 𝑅 two, and we can label the potential
difference across resistor 𝑅 three as 𝑉 𝑅 three. Then by applying Kirchhoff’s second
law to loop one, we find that 𝑉 two minus 𝑉 𝑅 two minus 𝑉 𝑅 three equals
zero. Notice that the potential
difference provided by the cell 𝑉 two is positive and the potential difference
across each resistor is negative. We can get expressions for 𝑉 𝑅
two and 𝑉 𝑅 three by using Ohm’s law.
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. The current in resistor 𝑅 two is
𝐼 two, so the potential difference across resistor 𝑅 two is 𝑉 𝑅 two equals 𝐼
two 𝑅 two. Similarly, the current in resistor
𝑅 three is 𝐼 one, so the potential difference across resistor 𝑅 three will be 𝑉
𝑅 three equals 𝐼 one 𝑅 three.
So by substituting these equations
into the equation we obtained from Kirchhoff’s second law for loop one, we have the
equation 𝑉 two minus 𝐼 two 𝑅 two minus 𝐼 one 𝑅 three equals zero. From the diagram, we know that the
value of 𝑉 two is 18 volts. We are also given the currents 𝐼
one equals one amp and 𝐼 two equals three amps. And the resistances 𝑅 two equals
5.2 ohms and 𝑅 three equals 𝑅 ohms. By substituting these values into
the equation, we get 18 volts minus three amps times 5.2 ohms minus one amp times 𝑅
ohms equals zero. Multiplying through these brackets,
we find 18 minus 15.6 minus 𝑅 equals zero, which simplifies to 2.4 minus 𝑅 equals
zero. By adding 𝑅 to both sides of the
equation, we find that 𝑅 equals 2.4 ohms.
We have found the value of the
resistance 𝑅, and this answer corresponds with option (A). The value of the resistance 𝑅 is
𝑅 equals 2.4 ohms.