Compute the equivalent resistance of the network in Fig. 26.42, and find the current in each res

step 1 We're given the following circuit. We have a 60 -bobob battery attached to two parallel systems,

Compute the equivalent resistance of the network in Fig....

Key Concepts

Ohm's Law

Ohm's Law relates voltage (V), current (I), and resistance (R) through the equation V = IR. It provides the basic framework needed to calculate the current flowing through individual resistors once the effective voltage across them is determined.

Kirchhoff's Circuit Laws

Kirchhoff's laws, including the Current Law (which states that the total current entering a junction must equal the total current leaving it) and the Voltage Law (which states that the sum of the voltage drops around a closed loop must equal the total supplied voltage), are essential for analyzing and solving circuits, especially those that are not reducible to simple series or parallel combinations.

Equivalent Resistance

In circuit analysis, equivalent resistance is the single resistance that can replace a network of resistors without changing the overall current and voltage. This simplification makes it easier to analyze the circuit and determine the behavior of the system when connected to a voltage source.

Series and Parallel Resistors

Resistors in a network can be configured in series or parallel. In series, the resistances add together directly, increasing the total resistance. In parallel, the reciprocals of the resistances add, resulting in a total resistance that is less than the smallest individual resistance. Understanding these configurations is fundamental to reducing complex resistor networks.

Transcript

00:02 We're given the following circuit.

00:06 We have a 60 -bobob battery attached to two parallel systems, each with two resistors, r1 through r4.

00:26 Their values equaling 3 oms, 6 oms, 12 oms, and 4 oms.

00:46 We approach this circuit by simplifying it to the following.

01:02 We must solve for the equivalent resistance.

01:06 To solve for equivalent resistance, we use the...

01:09 Two following formulas, meaning that in the parallel systems of resistors, we'll add 1 over r plus r1 plus 1 over r2 until we've added all of them together, and then we'll take the inverse of that.

01:59 In series systems, we add them r1 plus r2 plus so on.

02:09 So we'll first look at the two systems in parallel, r1 and r2.

02:30 These are in parallel, so we add them as so.

02:49 Similarly, we'll add r3 and r4 together, and we'll assign these to r5 and r6.

03:23 So we can look at this circuit as this, and now we can add these together in series to find that the equivalent resistance is 5 oms.

03:59 So now you might wonder why do we need to solve for the equivalent resistance? this allows us to find the current here at these two points.

04:15 That's the current surrounding everything in the circuit.

04:24 This allows us to write oms law v equals ir, finding that the current in the circuit is 12 amps.

04:52 Now let's return to our original circuit.

05:08 We know that the current here is 12 volts, 12 amps, i'm sorry.

05:26 And so we'll write r1, 2, 3, and 4 as following.

05:39 To solve for each resistor's current, the current across each resistor, first we have to solve for the current entering and exiting each system.

05:56 And we know that series resistors hold the same current...

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