Series Circuit Formula

The Formula

R_{\text{total}} = R_1 + R_2 + R_3 + \ldots (resistances add up)

When to use: Like cars on a single-lane road — every car (charge) must pass through every toll booth (component) in order.

Quick Example

Old-style Christmas lights in series: one burns out and they all go dark because the circuit is broken.

Notation

R_{\text{eq}} is the total (equivalent) resistance in ohms (\Omega), R_i is the resistance of the i-th component, I is the common current in amperes, and V_i = IR_i is the voltage drop across each component.

What This Formula Means

A circuit arrangement in which components are connected end-to-end along a single path, so exactly the same current flows through every component.

Like cars on a single-lane road — every car (charge) must pass through every toll booth (component) in order.

Formal View

For n resistors in series carrying common current I, the equivalent resistance is R_{\text{eq}} = \sum_{i=1}^{n} R_i. By Kirchhoff's voltage law, V_{\text{source}} = \sum_{i=1}^{n} V_i = I \sum_{i=1}^{n} R_i.

Worked Examples

Example 1

medium
Three resistors (4 \text{ } \Omega, 6 \text{ } \Omega, 10 \text{ } \Omega) are connected in series to a 20 \text{ V} battery. What is the current and the voltage across each resistor?

Solution

  1. 1
    Total resistance: R_T = 4 + 6 + 10 = 20 \text{ } \Omega.
  2. 2
    Current (same through all): I = \frac{V}{R_T} = \frac{20}{20} = 1 \text{ A}
  3. 3
    Voltage drops: V_1 = IR_1 = 4 \text{ V}, V_2 = IR_2 = 6 \text{ V}, V_3 = IR_3 = 10 \text{ V}.
  4. 4
    Check: 4 + 6 + 10 = 20 \text{ V} (matches battery voltage).

Answer

I = 1 \text{ A}; \quad V_1 = 4 \text{ V}, V_2 = 6 \text{ V}, V_3 = 10 \text{ V}
In a series circuit, current is the same through all components. Resistances add directly, and the battery voltage divides among the resistors proportionally to their resistances.

Example 2

hard
A 6 \text{ V} battery powers a series circuit with a 3 \text{ } \Omega resistor and an unknown resistor. The current is 0.5 \text{ A}. Find the unknown resistance and the power dissipated by each resistor.

Common Mistakes

  • Thinking the current decreases as it passes through each resistor — the current is identical at every point in a series circuit; it is the voltage that drops across each component.
  • Using the parallel resistance formula (1/R) for series components — in series, resistances simply add: R_{\text{total}} = R_1 + R_2 + \ldots
  • Forgetting that if one component breaks (open circuit), the entire series circuit stops — there is only one current path, so a break anywhere halts all current flow.

Why This Formula Matters

Series circuits appear in voltage dividers, sensor circuits, string lights, and battery packs. Understanding series connections is essential for predicting how adding components affects current, for designing simple sensor readout circuits, and for troubleshooting faults where one broken component stops the whole circuit.

Frequently Asked Questions

What is the Series Circuit formula?

A circuit arrangement in which components are connected end-to-end along a single path, so exactly the same current flows through every component.

How do you use the Series Circuit formula?

Like cars on a single-lane road — every car (charge) must pass through every toll booth (component) in order.

What do the symbols mean in the Series Circuit formula?

R_{\text{eq}} is the total (equivalent) resistance in ohms (\Omega), R_i is the resistance of the i-th component, I is the common current in amperes, and V_i = IR_i is the voltage drop across each component.

Why is the Series Circuit formula important in Physics?

Series circuits appear in voltage dividers, sensor circuits, string lights, and battery packs. Understanding series connections is essential for predicting how adding components affects current, for designing simple sensor readout circuits, and for troubleshooting faults where one broken component stops the whole circuit.

What do students get wrong about Series Circuit?

Adding more resistors in series increases total resistance and decreases current.

What should I learn before the Series Circuit formula?

Before studying the Series Circuit formula, you should understand: circuit, resistance, ohms law.

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