Motor effect Equation, Derivation & Rearrangements

Core idea

Overview

The motor effect describes the physical force exerted on a current-carrying conductor when it is placed within an external magnetic field. This force arises from the interaction between the magnetic field generated by the moving charges in the wire and the surrounding external magnetic field.

When to use: Apply this equation when calculating the magnetic force on a straight conductor of length L carrying a steady current I within a uniform magnetic field B. It is specifically used when the conductor is rigid and the orientation relative to the field is known.

Why it matters: This principle is the core mechanism behind electric motors, which convert electrical energy into kinetic energy. It is also essential for the design of analog meters, loudspeakers, and electromagnetic particle accelerators.

Symbols

Variables

B = Magnetic Flux Density, I = Current, L = Length, F = Force

B

Magnetic Flux Density

T

I

Current

A

L

Length

m

F

Force

N

Walkthrough

Derivation

Understanding the Motor Effect

Describes the phenomenon where a current-carrying wire in a magnetic field experiences a force.

The field and current are uniform and constant.

1

State the Formula:

This gives the force magnitude. Direction is given by Fleming’s left-hand rule.

F = B I l sin θ

Result

F = B I l sin θ

Source: Standard curriculum — A-Level Physics

Free formulas

Rearrangements

Solve for $F$

Make F the subject

F = B I L sin θ

F is already the subject of the formula.

Difficulty: 1/5

Solve for $B$

Make B the subject

B = \frac{F}{I L sin θ}

Start from Motor effect. To make B the subject, divide by ILsintheta.

Difficulty: 2/5

Solve for $I$

Make I the subject

I = \frac{F}{B L sin θ}

Start from Motor effect. To make I the subject, divide by BLsintheta.

Difficulty: 2/5

Solve for $L$

Make L the subject

L = \frac{F}{B I sin θ}

Start from Motor effect. To make L the subject, divide by BIsintheta.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin, showing that Force is directly proportional to Current. For a physics student, this means that larger current values result in a proportionally larger force, while smaller current values lead to a weaker force. The most important feature is that the linear relationship means doubling the current will always double the force, provided the magnetic field, length, and angle remain constant.

Graph type: linear

Why it behaves this way

Intuition

Imagine the magnetic field lines from the external magnet pushing against the magnetic field lines created by the current in the wire, resulting in a net sideways push on the wire.

F

The magnetic force exerted on the current-carrying conductor

It's the push or pull that the magnetic field applies to the current-carrying wire.

B

Magnetic field strength (or magnetic flux density)

A measure of how strong the external magnetic field is; denser field lines mean a stronger field.

I

Electric current flowing through the conductor

The rate at which electric charge moves through the wire; more charge moving faster means a larger current.

L

Length of the conductor segment within the magnetic field

How much of the current-carrying wire is actually exposed to the magnetic field.

sin θ

The sine of the angle (θ) between the direction of the current and the direction of the magnetic field

Only the component of the magnetic field perpendicular to the current contributes to the force; if current is parallel to the field, there's no force (sin(0)=0).

Free study cues

Insight

Canonical usage

The equation is primarily used with SI units, where force is in Newtons, magnetic field strength in Teslas, current in Amperes, and length in meters.

Common confusion

A common mistake is using inconsistent units, such as length in centimeters while other quantities are in SI units, or incorrectly identifying the angle θ (it is the angle between the current direction and the magnetic

Dimension note

The `sin(θ)` term is dimensionless, as it represents the sine of an angle. While angles can be expressed in degrees or radians, the sine function itself yields a pure number.

Unit systems

$F$ N · Resultant magnetic force on the conductor.

$B$ T · Magnitude of the external magnetic field (magnetic flux density).

$I$ A · Magnitude of the steady current flowing through the conductor.

$L$ m · Length of the conductor segment within the magnetic field.

$θ$ rad · Angle between the direction of the current vector and the magnetic field vector. The sine function makes this term dimensionless.

Ballpark figures

Quantity:

One free problem

Practice Problem

A straight wire of length 0.8 meters carries a current of 5.0 A. It is positioned perpendicular (90°) to a uniform magnetic field of 0.2 T. Calculate the magnetic force acting on the wire.

Magnetic Flux Density0.2 T

Current5 A

Length0.8 m

Solve for: $F$

Hint: Since the wire is perpendicular, sin(90°) equals 1, meaning the force is simply the product of B, I, and L.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Predicting force on a motor armature.

Study smarter

Tips

Ensure all units are in SI: Tesla (B), Amperes (I), and Meters (L).
Use Fleming's Left-Hand Rule to determine the direction of the resulting force.
The force is maximized when the wire is perpendicular to the field and zero when parallel.
Always check if the angle θ is provided; if the wire is perpendicular, sin(θ) equals 1.

Avoid these traps

Common Mistakes

Mixing up direction rules.
Using charge formula instead of current.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Describes the phenomenon where a current-carrying wire in a magnetic field experiences a force.

Apply this equation when calculating the magnetic force on a straight conductor of length L carrying a steady current I within a uniform magnetic field B. It is specifically used when the conductor is rigid and the orientation relative to the field is known.

This principle is the core mechanism behind electric motors, which convert electrical energy into kinetic energy. It is also essential for the design of analog meters, loudspeakers, and electromagnetic particle accelerators.

Mixing up direction rules. Using charge formula instead of current.

Predicting force on a motor armature.

Ensure all units are in SI: Tesla (B), Amperes (I), and Meters (L). Use Fleming's Left-Hand Rule to determine the direction of the resulting force. The force is maximized when the wire is perpendicular to the field and zero when parallel. Always check if the angle θ is provided; if the wire is perpendicular, sin(θ) equals 1.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics
Wikipedia: Lorentz force
Britannica: Lorentz force
Wikipedia article 'Lorentz force'
Halliday, Resnick, Walker, Fundamentals of Physics, 10th Edition
Wikipedia: Earth's magnetic field
Wikipedia: Mains electricity
Standard curriculum — A-Level Physics

Motor effect

Overview

Variables

Derivation

State the Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Magnetic force

Frequently Asked Questions

Sources