Answer to Question #154944 in Electric Circuits for Jane

Question

 By considering the principles that the distribution of current in a circuit is the result of charge conservation, and the distribution of potential differences in a circuit is the result of energy conservation, derive:

(a) the combined resistance of two resistors R1 and R2 connected in series

(b) the combined resistance of two resistors R1 and R2 connected in parallel

Answer 1

To find the combined resistance of two resistors $R_1$ and $R_2$ connected in series or in parallel we can apply two Kirchhoff’s Rules: The Loop Rule and The Junction Rule. These rules are the consequences of two laws of conservation: The Law of Conservation of Energy and The Law of Conservation of Charge.

(a) Let's consider two resistors $R_1$ and $R_2$ connected in series. Since there is the only one path for the charges to flow through such circuit, the current will be same through each resistors. To find the voltage drop in the circuit we can apply the Loop Rule. The Loop Rule is the consequence of the law of Conservation of Energy. Since the charged particles travels around the circuit and returns to the same position where it is started, the net change in potential difference would be zero along the closed path. It can be written mathematically as follows:

$$\Delta V_{net}=\displaystyle\sum_{j}^n \Delta V_j=0,$$ $$V=V_1+V_2,$$ $$V=IR_1+IR_2=I(R_1+R_2).$$

From this equation we can find the current flowing through the circuit:

$$I=\dfrac{V}{(R_1+R_2)}=\dfrac{V}{R_{eq}}.$$

Because the current through each resistor is the same, we can substitute the sum of the two resistors by the combined (or equivalent) resistance. Therefore, the combined (or equivalent) resistance of two resistors $R_1$ and $R_2$ connected in series is the sum of the combination of the resistors:

$$R_{eq}=R_1+R_2.$$

b) Let's consider two resistors $R_1$ and $R_2$ connected in parallel. In this case, the voltage drop across each resistor is the same. To find the current in the circuit we can apply the Junction Rule. The Junction Rule is the consequence of the law of Conservation of Charge. According to it, the amount of charge can’t change. Therefore, the amount of current entering a junction would be equal to the amount of current leaving that junction. It can be written mathematically as follows:

$$\displaystyle\sum_{j}^n i_j = \displaystyle\sum_{k}^m i_k,$$ $$I=I_1+I_2.$$

Applying Ohm's Law, we get:

$$I=\dfrac{V_1}{R_1}+\dfrac{V_2}{R_2}=\dfrac{V}{R_1}+\dfrac{V}{R_2},$$ $$I=V(\dfrac{1}{R_1}+\dfrac{1}{R_2})=\dfrac{V}{R_{eq}}.$$

Finally, we can find the combined (or equivalent) resistance of two resistors $R_1$ and $R_2$connected in parallel:

$$R_{eq}=\dfrac{1}{(\dfrac{1}{R_1}+\dfrac{1}{R_2})}.$$

Answer:

(a) $R_{eq}=R_1+R_2.$

(b) $R_{eq}=\dfrac{1}{(\dfrac{1}{R_1}+\dfrac{1}{R_2})}.$