Inductor Voltage Relation: Voltage From Changing Current

This guide sits in the magnetic devices-and-networks lane of the Electromagnetism Principle Map, after Inductance-Flux Relation. The surrounding decisions are choosing the current reference direction, choosing the voltage polarity, and deciding whether the current is increasing or decreasing. Those choices support the relation; they are not separate principle keys.

On this page: The Principle | Conditions | Misconceptions | Elaborative Encoding | Retrieval Practice | Worked Example | Solve a Problem | Related Principles | FAQ | Related Guides | How This Fits

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The Principle

Statement

Inductor Voltage Relation states that the voltage across an inductor is proportional to the time rate of change of current through it. The inductance $L$ sets how much voltage is needed for a given current-change rate.

Mathematical Form

$\Delta V_L = L\frac{dI}{dt}$

Where:

• $\Delta V_L$ is the voltage across the inductor, in volts
• $L$ is the inductance, in henries
• $I$ is the current through the inductor, in amperes
• $\frac{dI}{dt}$ is the time rate of change of current, in amperes per second

Here $\Delta V_L$ means the potential difference across the inductor’s terminals, not a change in voltage over time; the time-change appears in $\frac{dI}{dt}$.

The diagram fixes the sign convention before the equation is used. Current is chosen to enter the terminal marked positive, so a positive $\frac{dI}{dt}$ gives a positive $\Delta V_L$ in that chosen voltage-reference polarity. If you reverse the voltage polarity or current reference, the sign bookkeeping changes.

Useful rearrangements

When the relation is valid, you can solve for current-change rate or inductance:

$\frac{dI}{dt}=\frac{\Delta V_L}{L}$

$L=\frac{\Delta V_L}{dI/dt}$

These are algebraic rearrangements of the same inductor model, not new principles.

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Conditions of Applicability

Condition: inductor model; passive sign convention fixed

Practical modeling notes

• Inductor model means the component or coil is being represented by an inductance $L$, rather than by a full magnetic-field calculation.
• Passive sign convention fixed means the current reference enters the terminal chosen as positive for $\Delta V_L$.
• In many introductory problems, $L$ is treated as constant. If the inductor saturates or changes geometry, the relation may need a more detailed model.
• If current is steady, $\frac{dI}{dt}=0$, so an ideal inductor has zero voltage across it even while current flows.

When it does not apply directly

• No inductor model: a generic magnetic field or coil geometry may need a flux or field relation first.
• Unfixed sign convention: the equation cannot tell you the sign until voltage polarity and current reference are chosen.
• Nonlinear inductance: if $L$ changes strongly with current, a single constant $L$ may not describe the whole interval.

Want the complete framework behind this guide? Read Masterful Learning.

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Common Misconceptions

Misconception 1: An inductor’s voltage depends on current itself

The truth: In the ideal constant-$L$ model, voltage depends on $\frac{dI}{dt}$, not on $I$ alone.

Why this matters: A large steady current can have zero inductor voltage, while a smaller current that is changing rapidly can require a large voltage.

Misconception 2: The sign comes from the inductor automatically

The truth: The sign comes from the chosen current direction and voltage polarity. Passive sign convention must be fixed before interpreting positive or negative voltage.

Why this matters: Without the convention, the same physical situation can be described with opposite signed variables.

Misconception 3: The relation is Ohm’s law for inductors

The truth: Ohm’s law relates voltage to current through resistance. Inductor Voltage Relation relates voltage to current-change rate through inductance.

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Elaborative Encoding

Use these questions to build understanding before memorizing the formula. See Elaborative Encoding for the broader method.

Within the Principle

• Why does the equation use $\frac{dI}{dt}$ instead of only $I$?
• What units must $L$ have so that $L\frac{dI}{dt}$ has units of volts?

For the Principle

• What words or circuit labels tell you that a lumped inductor model is intended?
• Before using the equation, how would you mark the voltage polarity and current direction so the sign is meaningful?

Between Principles

• How does this relation build on Inductance-Flux Relation without being a flux-linkage equation itself?

Generate an Example

• Describe a circuit interval where current through an inductor is constant. What voltage does the ideal inductor model predict?

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Retrieval Practice

Answer from memory, then reveal the result and check it. See Retrieval Practice for the full study method.

State the principle in words: _____For an inductor model with passive sign convention fixed, inductor voltage is proportional to the time rate of change of current.

Write the canonical equation: _____$\Delta V_L = L\frac{dI}{dt}$

State the canonical condition: _____inductor model; passive sign convention fixed

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Worked Example

Use this worked example to practice Self-Explanation.

Problem

An inductor has $L=0.20\,\mathrm{H}$. Current enters the terminal marked positive for $\Delta V_L$. During a linear ramp, the current changes from $I_i=0.50\,\mathrm{A}$ to $I_f=1.70\,\mathrm{A}$ in $\Delta t=0.030\,\mathrm{s}$. Find $\Delta V_L$ during the ramp.

Step 1: Verbal Decoding

Target: $\Delta V_L$
Given: $L$, $I_i$, $I_f$, $\Delta t$
Constraints: inductor model; passive sign convention fixed; current changes linearly

Step 2: Visual Decoding

Draw one inductor, mark the current entering the positive terminal, and label the voltage polarity across the same inductor. Mark the current ramp from $I_i$ to $I_f$. (The key visual fact is that current enters the positive terminal while the current value increases.)

Step 3: Physics Modeling

1. $\Delta V_L=L\frac{I_f-I_i}{\Delta t}$

Step 4: Mathematical Procedures

1. $\Delta V_L=(0.20\,\mathrm{H})\frac{1.70\,\mathrm{A}-0.50\,\mathrm{A}}{0.030\,\mathrm{s}}$
2. $\Delta V_L=(0.20\,\mathrm{H})(40\,\mathrm{A/s})$
3. $\underline{\Delta V_L=8.0\,\mathrm{V}}$

Step 5: Reflection

• Dimensional analysis: A henry times amperes per second gives volts.
• Interpretation: The positive result matches the chosen polarity because current enters the positive terminal and is increasing.
• Limiting case: If the current were constant over the same interval, the model would give $\Delta V_L=0$.

Before moving on: self-explain the model

Try explaining why Step 3 uses current-change rate, why the sign convention matters, and why the answer is positive for this ramp.

Physics model with explanation

Principle: We use Inductor Voltage Relation because the problem gives an inductor model, an inductance, and a changing current.

Conditions: The passive sign convention is fixed because current is stated to enter the terminal marked positive for $\Delta V_L$.

Relevance: The target is the inductor voltage during a linear current ramp, so the current derivative is the needed model input.

Description: A linear current ramp has constant slope. Multiplying that slope by $L$ gives the voltage in the chosen polarity.

Goal: Substitute the current-change rate into the inductor voltage relation and keep the sign attached to the chosen convention.

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Solve a Problem

Apply what you have learned with Problem Solving.

Problem

An inductor has $L=0.50\,\mathrm{H}$. Current enters the terminal marked positive for $\Delta V_L$. For $\Delta t=0.20\,\mathrm{s}$, the measured inductor voltage is $\Delta V_L=-3.0\,\mathrm{V}$. Find the current change $\Delta I$ over that interval.

Hint: A negative voltage in the chosen polarity means the current is decreasing.

Show Solution

Step 1: Verbal Decoding

Target: $\Delta I$
Given: $L$, $\Delta V_L$, $\Delta t$
Constraints: inductor model; passive sign convention fixed; voltage is constant over the interval

Step 2: Visual Decoding

Draw one inductor, mark current entering the positive terminal, and label the voltage polarity across the same inductor. Note that the voltage label is negative in that polarity. (The key visual fact is that the chosen polarity stays fixed while the current decreases.)

Step 3: Physics Modeling

1. $\Delta V_L=L\frac{\Delta I}{\Delta t}$

Step 4: Mathematical Procedures

1. $\Delta I=\frac{\Delta V_L\Delta t}{L}$
2. $\Delta I=\frac{(-3.0\,\mathrm{V})(0.20\,\mathrm{s})}{0.50\,\mathrm{H}}$
3. $\underline{\Delta I=-1.2\,\mathrm{A}}$

Step 5: Reflection

• Dimensional analysis: Volt-seconds divided by henries gives amperes.
• Verification: Substituting $\Delta I=-1.2\,\mathrm{A}$ gives $\Delta V_L=-3.0\,\mathrm{V}$.
• Interpretation: The negative sign means the current decreased relative to the chosen current direction.

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Related Principles

See Electromagnetism: The Principle Map for where inductors sit in the magnetic devices-and-networks lane.

Principle	Relationship to Inductor Voltage Relation
Inductance-Flux Relation	Connects current to magnetic flux linkage before current changes create inductor voltage.
Kirchhoff Loop Rule	Supplies the loop-voltage bookkeeping that often surrounds inductor voltage in circuits.
Inductor Energy	Later uses inductance and current to model magnetic-field energy stored in an inductor.

See Principle Structures for a broader view of how one equation can depend on setup decisions such as signs and reference directions.

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FAQ

What is the inductor voltage relation?

The relation is $\Delta V_L=L\frac{dI}{dt}$. It says voltage across an inductor is proportional to the rate at which current through the inductor changes.

When does the inductor voltage relation apply?

It applies under the canonical condition: inductor model; passive sign convention fixed. The setup must represent the component by an inductance and define current direction and voltage polarity.

What does passive sign convention mean for an inductor?

Passive sign convention means the chosen current reference enters the terminal chosen as positive for the voltage variable. With that convention, $\Delta V_L=L\frac{dI}{dt}$ uses a positive voltage when current is increasing.

Why is voltage zero when inductor current is constant?

If current is constant, then $\frac{dI}{dt}=0$. The ideal inductor model therefore predicts $\Delta V_L=0$ even though current may still be flowing.

Is the inductor voltage relation the same as Ohm’s law?

No. Ohm’s Law relates voltage to current through resistance, while Inductor Voltage Relation relates voltage to current-change rate through inductance.

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Related Guides

• Electromagnetism: The Principle Map - Place inductors in the broader EM structure.
• Inductance-Flux Relation - Review the current-to-flux-linkage relation that comes before this guide.
• Kirchhoff Loop Rule - Connect inductor voltage to loop-voltage bookkeeping.
• Problem Solving - Practice translating givens, sign conventions, and diagrams into equations.

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How This Fits in Unisium

Unisium treats Inductor Voltage Relation as a principle because the equation is short but the setup is easy to misread. The useful learning path is to encode the passive sign convention, retrieve $\Delta V_L=L\frac{dI}{dt}$ with its exact condition, self-explain the current-change model, and solve new problems where the target or sign changes.

Ready to master Inductor Voltage Relation? Check access and join the Unisium waitlist or explore the full framework in Masterful Learning.

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