Solve Electric Circuit Equation: Use Kirchhoff & Ohm's Law

[butz3]

please help I'm am having lots of trouble undersanding electric circuits. my question says " use kirchhoff's loop rule and ohm's law to derive the equation for the equivalent resistance of four resistors connected in a series. concisely explain each step. " so i know that the eqaution for equivalent resistance is R=R1+R2+R3+R4 and that kirchhoff's loop rule is EV=0 and that ohm's law is I^2R. but how do i use that to get the equivalent I'm so confused. please help.

 

[cepheid]

Let's consider this case for a really simple circuit with one loop consisting of an ideal voltage source and your four resistors, all in series.

First let's look at Kirchoff's Loop Rule, what we usually call Kirchoff's Voltage Law (KVL) in my class. It states that the sum of the potential differences around the loop must be equal to zero. In this case, that means that the voltage across the source is equal to the sum of the potential drops across the four resistors:

$$V_s - (V_1 + V_2 + V_3 + V_4) = 0$$

Let's talk about the current in the circuit. This being a loop (all elements in series), the current must be the same everywhere in the circuit. Detectors at any two different points along the loop should measure the same rate of charge per second flowing past, otherwise charge is piling up somewhere, or charge is not conserved. Since we have a steady current, and charge is conserved, the current is constant and equal to $I$ everywhere.

Ohm's law tells us two things:

1. This total current $I$ in the circuit is equal to the voltage of the source divided by the total (or equivalent) resistance of the circuit:

$$I = \frac{V_s}{R_{eq}}$$

$$\therefore \ \ V_s = R_{eq}I$$

2. The potential difference (ie voltage drop) across each resistor is given by the current through it times its resistance:

$$V_1 = R_{1}I$$
$$V_2 = R_{2}I$$
$$V_3 = R_{3}I$$
$$V_4 = R_{4}I$$

Substitute the results of 1 and 2 (from Ohm's law) into the the very first equation (from KVL):

$$V_s - (V_1 + V_2 + V_3 + V_4) = 0$$

$$R_{eq}I - (R_{1}I + R_{2}I + R_{3}I + R_{4}I) = 0$$

Cancel out I and rearrange terms:

$$R_{eq} = R_{1} + R_{2} + R_{3} + R_{4}$$

That's the result we were trying to prove.

 

[wais]

To solve this electric circuit equation, we will use Kirchhoff's loop rule and Ohm's law. The first step is to understand the given information and the question. We are given a circuit with four resistors connected in series and we need to find the equivalent resistance of the circuit.

Step 1: Understanding the concept of series connection

In a series connection, the resistors are connected one after the other, so the current passing through each resistor is the same. This means that the total voltage applied to the circuit is divided among the resistors.

Step 2: Applying Kirchhoff's loop rule

Kirchhoff's loop rule states that the sum of the voltage drops in a closed loop in a circuit is equal to the sum of the voltage sources in that loop. In this case, the closed loop is the entire circuit and the voltage sources are the batteries or power sources.

Applying this rule, we get:

EV = 0

Where E is the total voltage applied to the circuit and V is the sum of the voltage drops across the four resistors.

Step 3: Using Ohm's law

Ohm's law states that the current flowing through a conductor is directly proportional to the voltage applied to it and inversely proportional to its resistance.

Using Ohm's law, we can write the equation for the voltage drop across each resistor as:

V1 = I × R1

V2 = I × R2

V3 = I × R3

V4 = I × R4

Where I is the current passing through the circuit and R1, R2, R3, and R4 are the resistances of the four resistors.

Step 4: Combining the equations

Now, we can substitute the values of V1, V2, V3, and V4 in the first equation from step 2.

EV = 0

E (V1 + V2 + V3 + V4) = 0

Substituting the values of V1, V2, V3, and V4 from step 3, we get:

E (I × R1 + I × R2 + I × R3 + I × R4) = 0

E (I × (R1 + R2 + R3 + R4)) = 0

Step 5: Solving for equivalent resistance

Finally, we can rearrange the equation to find the equivalent resistance