Magnetic force Equation, Derivation & Rearrangements

Core idea

Overview

The magnetic force formula F=BIL calculates the force exerted on a straight current-carrying conductor when placed inside a uniform magnetic field. This relationship assumes that the current is flowing perpendicular to the magnetic field lines to achieve maximum force.

When to use: Apply this equation when a straight wire is positioned within a constant magnetic field and you need to determine the mechanical force acting on it. It is primarily used when the angle between the current and the magnetic field is 90 degrees.

Why it matters: This principle is the core physical foundation behind the operation of electric motors, which drive everything from industrial machinery to household appliances. Understanding this force allows engineers to design precise control systems for loudspeakers, galvanometers, and magnetic levitation technologies.

Symbols

Variables

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

B

Magnetic Flux Density

T

I

Current

A

L

Length of Conductor

m

F

Force

N

Walkthrough

Derivation

Understanding Magnetic Force on a Wire

Calculates the force experienced by a current-carrying conductor placed in a magnetic field (motor effect).

The magnetic field is uniform.
The wire is straight within the field.

1

State the Formula:

Force depends on flux density B, current I, wire length l in the field, and angle $θ$ between wire and field.

F = B I l sin θ

Note: If $θ = 9 0^{\circ}$ , then $F = B I l$ . Direction via Fleming’s left-hand rule.

Result

F = B I l sin θ

Source: Edexcel A-Level Physics — Electric and Magnetic Fields

Free formulas

Rearrangements

Solve for $F$

Make F the subject

F = B I L

F is already the subject of the formula.

Difficulty: 1/5

Solve for $B$

Make B the subject

B = \frac{F}{I L}

Start from Magnetic force. To make B the subject, divide by IL.

Difficulty: 2/5

Solve for $I$

Make I the subject

I = \frac{F}{B L}

Start from Magnetic force. To make I the subject, divide by BL.

Difficulty: 2/5

Solve for $L$

Make L the subject

L = \frac{F}{B I}

Start from Magnetic force. To make L the subject, divide by BI.

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 because Force is directly proportional to Magnetic Flux Density when current and length remain constant. For a physics student, this linear relationship means that a larger Magnetic Flux Density results in a proportionally stronger force, while a smaller value leads to a weaker force. The most important feature is that doubling the Magnetic Flux Density exactly doubles the Force, demonstrating a constant rate of change defined by the product of current and len

Graph type: linear

Why it behaves this way

Intuition

Imagine a straight stream of electric charges (current) flowing through a wire. When this wire cuts across invisible magnetic field lines perpendicularly, each moving charge experiences a tiny sideways push or pull, and

F

The magnitude of the mechanical force (push or pull) exerted on the conductor.

This is the resulting physical interaction; a larger F means a stronger push or pull on the wire.

B

The magnitude of the magnetic flux density, representing the strength of the external magnetic field.

A stronger magnetic field (larger B) means more intense interaction with the moving charges, leading to a greater force.

I

The magnitude of the electric current flowing through the conductor.

A larger current (more charge flow per second) means more charges are interacting with the magnetic field simultaneously, resulting in a greater force.

L

The length of the conductor segment that is immersed within and perpendicular to the magnetic field.

A longer segment of wire (larger L) means more moving charges are interacting with the magnetic field over a greater distance, contributing to a larger total force.

Free study cues

Insight

Canonical usage

This equation is normally used with all quantities expressed in consistent SI units to ensure the calculated force is in Newtons.

Common confusion

A common mistake is using non-SI units, such as Gauss for magnetic field strength or centimeters for length, without proper conversion to Tesla and meters, respectively. This leads to incorrect force values.

Unit systems

$F$ Newton (N) · The SI derived unit for force, equivalent to kg·m·s-2.

$B$ Tesla (T) · The SI derived unit for magnetic flux density, equivalent to N·A-1·m-1 or kg·s-2·A-1.

$I$ Ampere (A) · The SI base unit for electric current.

$L$ Meter (m) · The SI base unit for length.

Ballpark figures

Quantity:

One free problem

Practice Problem

A straight wire with a length of 0.5 meters carries a steady current of 4.0 Amperes. If it is placed in a uniform magnetic field of 0.2 Teslas perpendicular to the field lines, calculate the magnitude of the magnetic force acting on the wire.

Magnetic Flux Density0.2 T

Current4 A

Length of Conductor0.5 m

Solve for: $F$

Hint: Multiply the magnetic field strength by the current and the length of the wire segment.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating force on a coil in a motor.

Study smarter

Tips

Ensure the magnetic flux density (B) is measured in Tesla (T).
The length (L) should only include the portion of the wire that is actually inside the magnetic field.
Use the Right-Hand Rule to determine the direction of the force relative to the current and field.

Avoid these traps

Common Mistakes

Using the charge formula instead of current.
Forgetting length is in meters.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Calculates the force experienced by a current-carrying conductor placed in a magnetic field (motor effect).

Apply this equation when a straight wire is positioned within a constant magnetic field and you need to determine the mechanical force acting on it. It is primarily used when the angle between the current and the magnetic field is 90 degrees.

This principle is the core physical foundation behind the operation of electric motors, which drive everything from industrial machinery to household appliances. Understanding this force allows engineers to design precise control systems for loudspeakers, galvanometers, and magnetic levitation technologies.

Using the charge formula instead of current. Forgetting length is in meters.

Estimating force on a coil in a motor.

Ensure the magnetic flux density (B) is measured in Tesla (T). The length (L) should only include the portion of the wire that is actually inside the magnetic field. Use the Right-Hand Rule to determine the direction of the force relative to the current and field.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics
Wikipedia: Magnetic force
Britannica: Magnetic force
NIST CODATA
IUPAC Gold Book
Wikipedia: Tesla (unit)
Wikipedia: Newton (unit)
Halliday, Resnick, Walker, Fundamentals of Physics, 10th Edition

Magnetic force

Overview

Variables

Derivation

State the Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Motor effect

Frequently Asked Questions

Sources