Magnetic Force On A Moving Charge: Use the Cross Product

This guide follows the electric field and circuit branches in the Electromagnetism Principle Map and starts the magnetic field-force lane. The surrounding decisions are right-hand-rule orientation, page-direction convention, vector component setup, and charge-sign interpretation. Those choices are setup work 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 Moving Charge gives the force on a charge moving through a magnetic field. The vector form $\vec{F}=q\vec{v}\times\vec{B}$ carries both size and direction: $\vec{v}\times\vec{B}$ gives the perpendicular direction for a positive charge, and a negative charge reverses it.

Mathematical Form

$\vec{F} = q\vec{v} \times \vec{B}$

Where:

• $\vec{F}$ is the magnetic force on the charge, in newtons
• $q$ is the signed charge, in coulombs
• $\vec{v}$ is the charge velocity, in meters per second
• $\vec{B}$ is the magnetic field, in tesla

The diagram is a guide-level orientation scaffold, not a substitute for learning to draw the setup in problems. It shows one standard case: a positive charge moves right while $\vec{B}$ points into the page, so $q\vec{v}\times\vec{B}$ points upward.

Magnitude form

When you only need the size of the force, use the accepted magnitude form:

$F = |q|vB\sin\theta$

Here $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. This scalar form gives zero force when velocity is parallel to the magnetic field and maximum force when velocity is perpendicular to the field. It does not by itself give the force direction.

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Conditions of Applicability

Condition: moving charge in a magnetic field

Practical modeling notes

• Moving charge means the velocity in the coordinate frame where $\vec{B}$ is specified matters; a stationary charge has no magnetic-force term from this relation.
• Magnetic field means $\vec{B}$ is known or modeled at the charge’s location.
• The sign of $q$ is part of the vector equation. Use the right-hand rule for $\vec{v}\times\vec{B}$, then reverse the direction if the charge is negative.
• If an electric field also acts on the charge, this magnetic-force term becomes part of the later Lorentz force relation.

When it does not apply directly

• Charge at rest: if $v=0$, the magnetic force from this relation is zero.
• Velocity parallel to field: if $\theta=0^\circ$ or $180^\circ$, the magnetic force is zero even though the charge is moving.
• Electric and magnetic fields together: use the combined Lorentz force when the total electromagnetic force is needed.

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

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Common Misconceptions

Misconception 1: A magnetic field pushes every charge

The truth: The magnetic part of the force depends on velocity. A charge at rest in a magnetic field does not feel this magnetic force.

Why this matters: Problems often include both electric and magnetic fields; the magnetic term only belongs when the charge is moving.

Misconception 2: The force points with the magnetic field

The truth: Magnetic force is perpendicular to both $\vec{v}$ and $\vec{B}$, not along $\vec{B}$.

Why this matters: Direction errors usually come from treating the magnetic field like an electric field instead of using the cross product.

Misconception 3: Charge sign affects only magnitude

The truth: The magnitude uses $|q|$, but the vector force uses the signed charge $q$. Negative charge reverses the force 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 a cross product rather than ordinary multiplication?
• What does the signed charge $q$ change that $|q|$ in the magnitude form does not?

For the Principle

• What wording in a problem tells you the charge is moving through the magnetic field rather than sitting in it?
• Before deciding direction, what orientation convention must be clear about the page, axes, or component basis?

Between Principles

• How is this different from Electric Field-Force Relation, where the force can point with or against the field depending only on charge sign?

Generate an Example

• Describe a moving-charge situation where changing only the sign of the charge reverses the magnetic force direction.

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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 moving charge in a magnetic field feels a force equal to its signed charge times velocity cross magnetic field.

Write the canonical equation: _____$\vec{F} = q\vec{v} \times \vec{B}$

State the canonical condition: _____moving charge in a magnetic field

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Worked Example

Use this worked example to practice Self-Explanation.

Problem

A proton with charge $q=+1.60\times10^{-19}\,\text{C}$ moves in the $+\hat{x}$ direction at speed $v=2.0\times10^6\,\text{m/s}$. It enters a uniform magnetic field of magnitude $B=0.40\,\text{T}$ pointing in the $-\hat{z}$ direction. Find the magnetic force vector on the proton.

Step 1: Verbal Decoding

Target: $\vec{F}$
Given: $q$, $v$, $B$
Constraints: charge is moving; magnetic field is known; velocity is perpendicular to field; proton has positive charge

Step 2: Visual Decoding

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

Step 3: Physics Modeling

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

Step 4: Mathematical Procedures

1. $\vec{F}=-qvB(\hat{x}\times\hat{z})$
2. $\vec{F}=qvB\hat{y}$
3. $\vec{F}=(1.60\times10^{-19}\,\text{C})(2.0\times10^6\,\text{m/s})(0.40\,\text{T})\hat{y}$
4. $\underline{\vec{F}=1.28\times10^{-13}\,\text{N}\,\hat{y}}$

Step 5: Reflection

• Dimensional analysis: $\text{C}\cdot\text{m/s}\cdot\text{T}$ gives newtons.
• Interpretation: The positive charge keeps the $\vec{v}\times\vec{B}$ direction, so the force points $+\hat{y}$.
• Limiting case: If the proton moved parallel to $\vec{B}$, the cross product would be zero.

Before moving on: self-explain the model

Try explaining why Step 3 uses the vector form, why the $-\hat{z}$ field changes the cross-product direction, and why the proton’s positive charge does not reverse that direction.

Physics model with explanation

Principle: We use Magnetic Force On A Moving Charge because the problem asks for the force on one moving charge in a magnetic field.

Conditions: The charge is moving and the magnetic field at the charge’s location is specified.

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

Description: Velocity points along $+\hat{x}$ and the magnetic field points along $-\hat{z}$. Their cross product points along $+\hat{y}$, and the positive charge keeps that direction.

Goal: Compute the cross-product direction and multiply the magnitudes to get the force vector.

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Solve a Problem

Apply what you have learned with Problem Solving.

Problem

An electron with charge $q=-1.60\times10^{-19}\,\text{C}$ moves in the $+\hat{x}$ direction at speed $v=3.0\times10^6\,\text{m/s}$. It enters a uniform magnetic field of magnitude $B=0.20\,\text{T}$ pointing in the $-\hat{z}$ direction. Find the magnetic force vector on the electron.

Hint: First find the $\vec{v}\times\vec{B}$ direction, then apply the negative charge sign.

Show Solution

Step 1: Verbal Decoding

Target: $\vec{F}$
Given: $q$, $v$, $B$
Constraints: charge is moving; magnetic field is known; velocity is perpendicular to field; electron has negative charge

Step 2: Visual Decoding

Draw $+\hat{x}$ to the right, $+\hat{y}$ upward, and $+\hat{z}$ out of the page. Draw $\vec{v}$ to the right and $\vec{B}$ into the page. (The key visual fact is that $\vec{v}\times\vec{B}$ points $+\hat{y}$ before the negative charge sign reverses it.)

Step 3: Physics Modeling

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

Step 4: Mathematical Procedures

1. $\vec{F}=e vB(\hat{x}\times\hat{z})$
2. $\vec{F}=-e vB\hat{y}$
3. $\vec{F}=-(1.60\times10^{-19}\,\text{C})(3.0\times10^6\,\text{m/s})(0.20\,\text{T})\hat{y}$
4. $\underline{\vec{F}=-9.6\times10^{-14}\,\text{N}\,\hat{y}}$

Step 5: Reflection

• Dimensional analysis: Charge times speed times magnetic field gives force units.
• Interpretation: The electron force points opposite the positive-charge cross-product direction.
• Verification: The force is perpendicular to both the velocity and the magnetic field.

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Related Principles

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

Principle	Relationship to Magnetic Force On A Moving Charge
Electric Field-Force Relation	Also gives force on a charge, but an electric field force is not velocity-cross-field dependent.
Lorentz Force	synthesis relation: combines electric force with this magnetic force term.
Magnetic Force On A Wire	current-carrying-wire analog: replaces the moving charge with current through a length vector.

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

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FAQ

What is the magnetic force on a moving charge?

The magnetic force on a moving charge is $\vec{F}=q\vec{v}\times\vec{B}$. It is the signed charge times the cross product of the charge velocity and the magnetic field.

When does magnetic force on a moving charge apply?

It applies under the canonical condition: moving charge in a magnetic field. If the charge is not moving, this magnetic-force term is zero.

Why is the magnetic force perpendicular to velocity?

The cross product $\vec{v}\times\vec{B}$ produces a vector perpendicular to both input vectors. Because the magnetic force is perpendicular to velocity, it changes the direction of motion but does no work on the charge.

What is the magnitude of magnetic force on a moving charge?

The magnitude is $F=|q|vB\sin\theta$, where $\theta$ is the angle between velocity and magnetic field. Use this form only for size; use the vector form for direction.

How does negative charge change magnetic-force direction?

The right-hand rule gives the direction of $\vec{v}\times\vec{B}$ for a positive charge. A negative charge multiplies that vector by a negative sign, so the force points the opposite way.

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Related Guides

• Electromagnetism: The Principle Map - Place this magnetic force law in the wider EM structure.
• Electric Field-Force Relation - Compare electric force with velocity-dependent magnetic force.
• Electric Field Superposition - Review vector addition before working with magnetic cross products.
• Problem Solving - Practice translating direction conventions into equations.

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How This Fits in Unisium

Unisium treats Magnetic Force On A Moving Charge as a principle because the equation is compact but the orientation work is easy to blur. The useful learning path is to encode the cross-product meaning, retrieve the vector and magnitude forms with the condition, self-explain charge-sign reversal, and solve new problems where the direction is not already decided.

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

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