Answer to Question #199868 in Electrical Engineering for richard gate

Question

i. Two or more resistors in a circuit are said to be in parallel when all the resistors are connected to the same nodes and the same voltage is appearing across all these elements. Two or more resistors in a circuit are also said to be in series when the current flowing through all the resistors is the same. With the aid of three resistors, Ohm’s law and diagrams derive the two important total resistance formulas for the two connections to verify these two postulates

Answer 1

Series

Because the voltage source's current passes through each resistor, the current through each resistor is the same. The current flowing through the circuit is affected by the voltage supplied by the voltage source as well as the resistance of the resistors. As a current flows through each resistor, a potential drop occurs that is equal to the loss of electric potential energy. The potential drop V across a resistor when a current passes through it is computed using Ohm's law, using the equation V=IR, where I is the current in amps (A) and R is the resistance in ohms. Because energy is conserved and voltage equals potential energy per charge, the sum of the voltage given to the circuit by the source and the potential drops across the different resistors in a loop should be zero

$\sum^N_{i=1}V_i=0$

The total of each resistor's potential drop and the voltage provided by the voltage source should equal zero:

$V-V_1-V_2-V_3=0$

$V=V_1+V_2+V_3 = IR_1+IR_2+IR_3$

$I=\frac{V}{R_1+R_2+R_3}=\frac{V}{R_{eq}}$

Because the current through each component is the same, the equivalence can be reduced to an equivalent resistance, which is just the sum of the individual resistors' resistances.

In series, any number of resistors can be linked. The equivalent resistance of N resistors linked in series is $V=R_1+R_2+R_3+...+R_N$

Parallel

Parallel resistors connected to a voltage source When one end of all the resistors is linked by a continuous wire of negligible resistance and the other end of all the resistors is likewise linked to one another by a continuous wire of negligible resistance, the resistors are in parallel. Each resistor has the same potential drop. The current through each resistor may be calculated using Ohm's rule I=VR, where the voltage across each resistor is constant. For example, an automobile's headlights, radio, and other systems are wired in parallel so that each subsystem may work totally independently while utilizing the entire voltage of the source.

The current flowing from the voltage source into the circuit is determined by the voltage supplied by the voltage source and the circuit's equivalent resistance. In this case, current flows from the voltage source to a junction, or node, where the circuit divides, passing through resistors R1 and R2. As charges flow from the battery, some pass through resistor R1 while others pass through resistance R2. The total of currents flowing into and out of a junction must equal the amount of currents flowing into and out of the junction:

$\sum I_{in}=\sum I_{out}$

$I=I_1+I_2=\frac{V_1}{R_1}+\frac{V_2}{R_2}=\frac{V}{R_1}+\frac{V}{R_2}=V(\frac{1}{R_1}+\frac{1}{R_2})=\frac{V}{R_{eqv}}$