Grade 10

Advanced

7 min

Lesson 5 of 12

Series Circuit Analysis

Not Started

Deep dives into the behavior of components connected in a single loop.

Why does a single blown bulb in an old string of holiday lights plunge the entire house into darkness, while modern sets stay lit?

Learning Objectives

1.Calculate the equivalent resistance ( $R_{eq}$ ) for multiple resistors connected in a single loop.
2.Apply the principle of conservation of energy to explain why voltage 'drops' across components.
3.Demonstrate why current ( $I$ ) remains identical at every point in a series circuit.

The Unbroken Stream: Constant Current

In a series circuit, components are connected end-to-end, forming a single path for electrons. Think of it like a single-lane circular road: every car (charge) must pass through every toll booth (resistor) in sequence. Because there are no branches, the current (

I

)—which is the rate of charge flow—cannot 'leak' out or pile up. Therefore, the current measured at any point in the loop is exactly the same. Mathematically, we express this as:

I_{total} = I_1 = I_2 = I_3 = ... = I_n

2A

of current leaves the battery,

2A

must pass through every single resistor before returning to the source.

Quick Check

If an ammeter placed before the first resistor in a series circuit reads $0.5 A$ , what will an ammeter placed after the third resistor read?

Remember the 'single-lane road' analogy.

Answer

$0.5 A$ , because current is constant at all points in a series circuit.

Resistance: The Cumulative Toll

Every component in a circuit offers some resistance to the flow of charge. In series, these resistances add up because the current must fight through each one sequentially. The Equivalent Resistance (

R_{eq}

) is the total resistance that the battery 'sees.' To find it, we simply sum the individual values:

R_{eq} = R_1 + R_2 + R_3 + ... + R_n

Adding more resistors in series always increases the total resistance, which in turn reduces the total current flowing through the circuit (assuming voltage stays the same).

A circuit contains a $9V$ battery and three resistors with values of $2 \Omega$ , $4 \Omega$ , and $6 \Omega$ connected in series.

1. Identify the formula: $R_{eq} = R_1 + R_2 + R_3$ 2. Substitute the values: $R_{eq} = 2 \Omega + 4 \Omega + 6 \Omega$ 3. Calculate the sum: $R_{eq} = 12 \Omega$

Voltage: The Energy Staircase

Voltage represents the electrical potential energy per unit charge. According to the Law of Conservation of Energy, the energy provided by the battery must be fully 'spent' by the time the charges return to the negative terminal. This is known as Kirchhoff’s Voltage Law (KVL). As current passes through each resistor, some energy is converted into heat or light, resulting in a voltage drop (

V

). The sum of these drops must equal the source voltage:

V_{source} = V_1 + V_2 + V_3 + ... + V_n

Using Ohm's Law (

V = IR

), we can see that the resistor with the highest resistance will cause the largest voltage drop.

Using the previous circuit ( $V_{source} = 9V$ , $R_{eq} = 12 \Omega$ ):

1. Find the total current: $I = \frac{V_{source}}{R_{eq}} = \frac{9V}{12 \Omega} = 0.75 A$ 2. Calculate the drop across the $4 \Omega$ resistor ( $V_2$ ): $V_2 = I \times R_2$ 3. $V_2 = 0.75 A \times 4 \Omega = 3V$ 4. This means $3V$ of the $9V$ total is 'used' by the second resistor.

Quick Check

If a $12V$ battery is connected to two identical resistors in series, what is the voltage drop across each one?

Conservation of energy says the sum must equal

12V

Answer

$6V$ each, because the total voltage is shared equally among identical resistances.

Advanced Application: The Voltage Divider

Series circuits are often used as voltage dividers. Since the current is constant, the ratio of the voltage drops is identical to the ratio of the resistances. If you need a specific voltage that is lower than your battery, you can use two resistors to 'tap' into the desired level. The voltage across

R_1

is given by:

V_1 = V_{source} \times \left( \frac{R_1}{R_{eq}} \right)

This principle is the foundation for volume knobs (potentiometers) and many sensor circuits.

A technician has a $24V$ power supply and a sensitive component with a resistance of $10 \Omega$ that will break if the current exceeds $0.8 A$ . What is the minimum additional resistance ( $R_{add}$ ) that must be placed in series to protect the component?

1. Calculate the minimum required $R_{eq}$ to limit current to $0.8 A$ : $R_{eq} = \frac{V}{I} = \frac{24V}{0.8 A} = 30 \Omega$ 2. Use the series resistance formula: $R_{eq} = R_{comp} + R_{add}$ 3. Solve for $R_{add}$ : $30 \Omega = 10 \Omega + R_{add}$ 4. $R_{add} = 20 \Omega$

Key Takeaways

1Current is constant everywhere in a series circuit ( $I_{total} = I_1 = I_2$ ).
2Equivalent resistance is the sum of all individual resistances ( $R_{eq} = \sum R_i$ ).
3Total voltage is equal to the sum of individual voltage drops ( $V_{total} = \sum V_i$ ), reflecting conservation of energy.
4The component with the highest resistance will have the largest voltage drop.

Quick Check

Multiple Choice

Three resistors ( $10 \Omega$ , $20 \Omega$ , $30 \Omega$ ) are in series. What is the equivalent resistance?

Multiple Choice

If you add a fourth resistor to a series circuit, what happens to the total current?

True/False

In a series circuit, the sum of the voltage drops across all resistors must equal the battery voltage.

What's Next?

Review Tomorrow

In 24 hours, try to sketch a 3-resistor series circuit and write down the three primary equations for $I$ , $R$ , and $V$ from memory.

Practice Activity

Find a household device that uses multiple batteries (like a flashlight). Observe how the batteries are placed end-to-end; this is a series connection designed to increase total voltage!

Browse

Browse

Series Circuit Analysis

Learning Objectives

The Unbroken Stream: Constant Current

Resistance: The Cumulative Toll

Voltage: The Energy Staircase

Advanced Application: The Voltage Divider

Key Takeaways

Quick Check

What's Next?

Series Circuit Analysis

Learning Objectives

The Unbroken Stream: Constant Current

Resistance: The Cumulative Toll

Voltage: The Energy Staircase

Advanced Application: The Voltage Divider

Key Takeaways

Quick Check

What's Next?