Answered: In a series circuit, is the total…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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In a series circuit, is the total resistance equal to the product, sum, difference, or quotient of the individual resistances?

Expert Solution

Step 1: Proof of sum of resistance in series combination

In a series circuit, the total resistance is equal to the sum of the individual resistances. This is a fundamental property of series circuits, and it can be explained in detail using Ohm's Law and the principles of electrical circuits.

Let's delve into the explanation step by step:

Ohm's Law: Ohm's Law describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit:
V = I * R
In this equation:
- V represents voltage (measured in volts, V).
- I represents current (measured in amperes, A).
- R represents resistance (measured in ohms, Ω).
Series Circuit: In a series circuit, multiple electrical components (such as resistors, bulbs, or any other devices) are connected end-to-end along a single pathway for current flow. In a series circuit:
- The same current flows through all components because there is only one path for the electrons to follow.
- The voltage source is shared across all components; the voltage is divided among them.
Voltage and Resistance in Series: When you have resistors connected in series, each resistor experiences the same current since they are in series. However, each resistor also has its own resistance, and Ohm's Law tells us that the voltage drop across a resistor is proportional to its resistance.
For each resistor (let's call them R1, R2, R3, and so on) in the series circuit, you can express the voltage drop (V_i) across it as:
$V subscript i space equals space I cross times R subscript i$
- V_i is the voltage drop across the i-th resistor.
- I is the current in the circuit (which is the same for all resistors).
- R_i is the resistance of the i-th resistor.
Total Voltage in Series: The total voltage (V_total) across the entire series circuit is equal to the sum of the voltage drops across all the individual resistors. Mathematically, you can express this as:
$V subscript t o t a l end subscript equals V subscript 1 plus V subscript 2 plus V subscript 3 plus...$
Using the expression for each resistor, you get:
$V subscript t o t a l end subscript equals I asterisk times R subscript 1 plus I asterisk times R subscript 2 plus I asterisk times R subscript 3 plus...$
Now, factor out the common current (I) from each term:
$V subscript t o t a l end subscript equals I cross times left parenthesis R subscript 1 plus R subscript 2 plus R subscript 3 plus... right parenthesis$
Total Resistance in Series: By comparing the total voltage ( $V subscript t o t a l end subscript$ ) to Ohm's Law (V = I * R), you can see that the total resistance (R_total) in a series circuit is directly proportional to the sum of the individual resistances:
$R subscript t o t a l end subscript equals R subscript 1 plus R subscript 2 plus R subscript 3 plus...$

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