Magnetic Force On A Wire: Formula and Direction

This guide follows Magnetic Force On A Moving Charge and Lorentz Force in the Electromagnetism Principle Map. The surrounding decisions are conventional-current direction, right-hand-rule orientation, page-direction convention, angle choice, and choosing vector versus magnitude form. Those choices are setup around the principle, 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

Magnetic Force On A Wire gives the force on a straight wire segment carrying current through a magnetic field. The vector form $\vec{F}=I\vec{L}\times\vec{B}$ uses a length vector $\vec{L}$ that points in the direction of conventional current, with $\vec{B}$ treated as the field across that segment, so the cross product carries the force direction.

Mathematical Form

$\vec{F} = I\vec{L} \times \vec{B}$

Where:

• $\vec{F}$ is the magnetic force on the wire segment, in newtons
• $I$ is the conventional current, in amperes
• $\vec{L}$ is the length vector of the straight segment, in meters, pointing with conventional current
• $\vec{B}$ is the magnetic field at the segment, in tesla

The diagram is a guide-level orientation scaffold. It shows one standard case: $I$ marks conventional current to the right, $\vec{L}$ points along the same straight segment, the magnetic field points into the page, and $I\vec{L}\times\vec{B}$ points upward.

Magnitude form

When direction is already known or not requested, use the scalar magnitude form:

$F = ILB\sin\theta$

Here $\theta$ is the angle between $\vec{L}$ and $\vec{B}$. The force is zero when the wire is parallel to the magnetic field and largest when the wire is perpendicular to the field. The scalar form gives size only; it does not decide the force direction.

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Conditions of Applicability

Condition: straight current-carrying segment in a magnetic field

Practical modeling notes

• Straight segment means one length vector $\vec{L}$ can represent the piece of wire you are modeling.
• Current-carrying means the direction of $\vec{L}$ follows conventional current, not electron drift.
• Magnetic field means $\vec{B}$ is known or treated as uniform enough over the segment for the chosen model.
• If the wire bends or the field varies along it, split the wire into smaller pieces or use the integral version later.

Special cases and extensions

• Wire parallel to field: the principle still applies, but the force is zero because $\sin\theta=0$.
• No current: if $I=0$, the relation gives zero magnetic force on the segment.
• Curved wire or varying field: a single $\vec{L}$ may hide changing directions or field changes, so model small wire elements or use the integral form.

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

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Common Misconceptions

Misconception 1: The force points with the magnetic field

The truth: The magnetic force is perpendicular to the current direction and the magnetic field direction.

Why this matters: Direction errors often come from treating magnetic force like electric force instead of using the cross product.

Misconception 2: The wire length is just a positive number

The truth: In the vector form, $\vec{L}$ is a direction-carrying length vector that points with conventional current.

Why this matters: Reversing current reverses $\vec{L}$ and therefore reverses the force direction.

Misconception 3: The magnitude formula is enough for every problem

The truth: $F=ILB\sin\theta$ gives the size of the force, but the vector form or a right-hand-rule setup is needed for direction.

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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 vector form use $\vec{L}$ instead of just the length $L$?
• What does $\sin\theta$ say about the force when the wire is parallel or perpendicular to the magnetic field?

For the Principle

• What wording in a problem tells you that a straight wire segment model is reasonable?
• Before deciding direction, what page or axis convention must be clear for the magnetic field?

Between Principles

• How is this current-carrying-wire force related to Magnetic Force On A Moving Charge?

Generate an Example

• Describe a wire-and-field setup where reversing the current reverses the magnetic force while the force size stays the same.

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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: _____A straight current-carrying wire segment in a magnetic field feels a force equal to current times the length vector cross the magnetic field.

Write the canonical equation: _____$\vec{F} = I\vec{L} \times \vec{B}$

State the canonical condition: _____straight current-carrying segment in a magnetic field

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Worked Example

Use this worked example to practice Self-Explanation.

Problem

A straight wire segment of length $L=0.40\,\text{m}$ carries conventional current $I=3.0\,\text{A}$ in the $+\hat{x}$ direction. The segment lies in a uniform magnetic field of magnitude $B=0.50\,\text{T}$ pointing in the $-\hat{z}$ direction. Find the magnetic force vector on the segment.

Step 1: Verbal Decoding

Target: $\vec{F}$
Given: $I$, $L$, $B$
Constraints: straight current-carrying segment; uniform magnetic field; length vector is perpendicular to magnetic field; conventional current points in the length-vector direction

Step 2: Visual Decoding

Draw $+\hat{x}$ to the right, $+\hat{y}$ upward, and $+\hat{z}$ out of the page. Draw the wire along $+\hat{x}$, mark conventional current to the right, and mark $\vec{B}$ into the page. (The key visual fact is that $\hat{x}\times(-\hat{z})=+\hat{y}$.)

Step 3: Physics Modeling

1. $\vec{F}=I(L\hat{x})\times(-B\hat{z})$

Step 4: Mathematical Procedures

1. $\vec{F}=-ILB(\hat{x}\times\hat{z})$
2. $\vec{F}=ILB\hat{y}$
3. $\vec{F}=(3.0\,\text{A})(0.40\,\text{m})(0.50\,\text{T})\hat{y}$
4. $\underline{\vec{F}=0.60\,\text{N}\,\hat{y}}$

Step 5: Reflection

• Dimensional analysis: Ampere times meter times tesla gives newtons.
• Interpretation: The force points upward because the current-length vector crossed into-page field gives $+\hat{y}$.
• Limiting case: If the wire were rotated parallel to the field, the magnetic force would be zero.

Before moving on: self-explain the model

Try explaining why Step 3 uses the vector form, why $\vec{L}$ points with conventional current, and why the force is perpendicular to the wire in this setup.

Physics model with explanation

Principle: We use Magnetic Force On A Wire because the problem asks for the force on a straight current-carrying segment in a magnetic field.

Conditions: The segment is straight, the current is specified, and the magnetic field is defined across the segment.

Relevance: The target is a force vector, so the vector form is the most direct model.

Description: Conventional current points along $+\hat{x}$, and the magnetic field points along $-\hat{z}$. Their cross product points along $+\hat{y}$.

Goal: Use the current-length vector and magnetic field direction to get the force vector.

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Solve a Problem

Apply what you have learned with Problem Solving.

Problem

A straight wire segment of length $L=0.25\,\text{m}$ carries conventional current $I=4.0\,\text{A}$ in the $+\hat{x}$ direction. The segment is in a uniform magnetic field of magnitude $B=0.30\,\text{T}$ pointing in the $+\hat{z}$ direction. Find the magnetic force vector on the segment.

Check the direction carefully: the magnetic field now points out of the page instead of into the page.

Show Solution

Step 1: Verbal Decoding

Target: $\vec{F}$
Given: $I$, $L$, $B$
Constraints: straight current-carrying segment; uniform magnetic field; length vector is perpendicular to magnetic field; conventional current points in the length-vector direction

Step 2: Visual Decoding

Draw $+\hat{x}$ to the right, $+\hat{y}$ upward, and $+\hat{z}$ out of the page. Draw the wire along $+\hat{x}$, mark conventional current to the right, and mark $\vec{B}$ out of the page. (The key visual fact is that $\hat{x}\times\hat{z}=-\hat{y}$.)

Step 3: Physics Modeling

1. $\vec{F}=I(L\hat{x})\times(B\hat{z})$

Step 4: Mathematical Procedures

1. $\vec{F}=ILB(\hat{x}\times\hat{z})$
2. $\vec{F}=-ILB\hat{y}$
3. $\vec{F}=-(4.0\,\text{A})(0.25\,\text{m})(0.30\,\text{T})\hat{y}$
4. $\underline{\vec{F}=-0.30\,\text{N}\,\hat{y}}$

Step 5: Reflection

• Dimensional analysis: Current times length times magnetic field gives force units.
• Interpretation: Reversing the magnetic field direction reverses the force direction from the worked example.
• Verification: The force is perpendicular to both the wire direction and the magnetic field.

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Related Principles

See Electromagnetism: The Principle Map for where this wire-force relation sits in the magnetic branch.

Principle	Relationship to Magnetic Force On A Wire
Magnetic Force On A Moving Charge	Microscopic analog: individual moving charges feel $q\vec{v}\times\vec{B}$.
Lorentz Force	General charge-force relation that combines electric and magnetic contributions.
Magnetic Field Near A Long Straight Wire	source relation: a current can also create a magnetic field around a wire.

See Principle Structures for a broader view of how force and field relations connect across a subdomain.

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FAQ

What is the magnetic force on a wire?

The magnetic force on a straight current-carrying wire segment is $\vec{F}=I\vec{L}\times\vec{B}$. The length vector points in the direction of conventional current.

When does Magnetic Force On A Wire apply?

It applies under the canonical condition: straight current-carrying segment in a magnetic field. If the wire is curved or the field changes along it, split the wire into smaller elements or use a more general model.

What is the magnitude of magnetic force on a wire?

The magnitude is $F=ILB\sin\theta$, where $\theta$ is the angle between the length vector and the magnetic field.

Which direction is the length vector?

The length vector $\vec{L}$ points along the straight segment in the direction of conventional current. It does not point with electron drift.

Why is the force zero when the wire is parallel to the field?

The cross product is zero when its input vectors are parallel or antiparallel. In the magnitude form, that same fact appears as $\sin 0^\circ=0$ or $\sin 180^\circ=0$.

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Related Guides

• Electromagnetism: The Principle Map - Place this force law in the wider EM structure.
• Magnetic Force On A Moving Charge - Compare wire force with the moving-charge version.
• Lorentz Force - See how electric and magnetic force terms combine for charges.
• Problem Solving - Practice translating direction conventions into equations.

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How This Fits in Unisium

Unisium treats Magnetic Force On A Wire as a principle because the equation is short but the representation is easy to mix up. The useful learning path is to encode what $\vec{L}$ means, retrieve the vector and magnitude forms with the condition, self-explain current direction and field orientation, and solve new problems where direction is not already decided.

Ready to master Magnetic Force On A Wire? Check access and join the Unisium waitlist or explore the full framework in Masterful Learning.

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