Derive the expressions for the equivalent resistence of two resistors R1 and R2 for the cases
when they are (a) in series and (b) in parallel. Hint: Set up the circuit with a constant Voltage
source V (it is not necessary to put in an exact value for V ). Define the current which leaves
(and returns to) the voltage source as I, and for the parallel circuit define the currents in each
leg of the circuit as I1 and I2. Write down Ohms law for each resistor, and also apply Kirchov’s
loop rule and juntion rule for each circuit. Manipulate and combine all these equations to obtain
an expression V = ReqI where Req is some function of R1 and R2 that you have determined. Req
is the equivalent resistance for the circuit.
This assignment is identical to the one millions before you have suffered thru. Follow the directions, and you will find a way through it.
Do you have any questions on it? It is rather straightforward as written.
To derive the expressions for the equivalent resistance of two resistors R1 and R2 when they are in series and parallel, we can follow the steps given in the hint. Let's start with the series configuration.
(a) Series Configuration:
In a series circuit, the resistors are connected one after the other, so the same current flows through both resistors. Let's assume the current leaving and returning to the voltage source is I.
1. Apply Ohm's Law to each resistor:
For resistor R1: V1 = R1 * I, where V1 is the voltage across R1.
For resistor R2: V2 = R2 * I, where V2 is the voltage across R2.
2. Apply Kirchhoff's Loop Rule:
The sum of the voltages around a closed loop is zero.
V = V1 + V2, where V is the applied voltage.
3. Substitute the voltage expressions from Step 1 into the Loop Rule equation:
V = (R1 * I) + (R2 * I)
V = (R1 + R2) * I
4. Rearrange the equation to solve for I:
I = V / (R1 + R2)
5. Rearrange the equation to solve for V:
V = I * (R1 + R2)
Comparing this equation with V = Req * I, we can see that the equivalent resistance Req for resistors in series is:
Req = R1 + R2
Now let's move on to the parallel configuration.
(b) Parallel Configuration:
In a parallel circuit, the resistors are connected side by side, so the voltage across both resistors is the same. Let's assume the currents through R1 and R2 are I1 and I2 respectively.
1. Apply Ohm's Law to each resistor:
For resistor R1: I1 = V / R1, where I1 is the current through R1.
For resistor R2: I2 = V / R2, where I2 is the current through R2.
2. Apply Kirchhoff's Junction Rule:
The sum of the currents entering a junction is equal to the sum of the currents leaving the junction.
I = I1 + I2, where I is the total current leaving and returning to the voltage source.
3. Substitute the current expressions from Step 1 into the Junction Rule equation:
I = (V / R1) + (V / R2)
4. Simplify the equation using a common denominator:
I = (VR2 + VR1) / (R1R2)
5. Rearrange the equation to solve for V:
V = I * (R1R2) / (R1 + R2)
Comparing this equation with V = Req * I, we can see that the equivalent resistance Req for resistors in parallel is:
1/Req = (1/R1) + (1/R2)
Or equivalently:
Req = (R1 * R2) / (R1 + R2)
So, the expressions for the equivalent resistance for resistors in series and parallel are:
(a) Series Configuration: Req = R1 + R2
(b) Parallel Configuration: Req = (R1 * R2) / (R1 + R2)