Magnetic Flux Density

What Is Magnetic Flux Density?

Magnetic flux density describes the strength and direction of a magnetic field at a particular point.

It is represented by the symbol:

B

Magnetic flux density is a vector quantity, which means it has both:

• magnitude
• direction

The direction of B is the direction shown by magnetic field lines at that point.

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The Tesla

The SI unit of magnetic flux density is the:

tesla (T)

Named after Nikola Tesla, the tesla is used to describe the strength of magnetic fields.

For example:

B = 0.20 T

means that the magnetic flux density is:

0.20 tesla

Smaller magnetic fields are often measured using:

millitesla (mT)

or:

microtesla (μT)

Conversions:

1 T = 1000 mT

1 mT = 10⁻³ T

1 μT = 10⁻⁶ T

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What Does a Larger B Mean?

A larger magnetic flux density means a stronger magnetic field.

For example:

B₁ = 0.80 T

B₂ = 0.20 T

Therefore:

B₁ > B₂

and the first magnetic field is stronger.

A stronger magnetic field can produce a larger force on a suitable:

• current-carrying conductor
• moving charge
• magnetic material
• magnet

under otherwise identical conditions.

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Magnetic Flux Density and Field Lines

Magnetic field diagrams can be used to represent differences in magnetic flux density.

In a consistently drawn diagram:

closer field lines → greater magnetic flux density

widely spaced field lines → smaller magnetic flux density

Therefore, field-line density provides a visual indication of relative magnetic field strength.

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Strong Fields Near Magnetic Poles

Around a bar magnet, magnetic field lines are usually most concentrated near the poles.

Therefore:

B is generally greatest near the poles

Farther from the magnet:

• field lines spread apart
• magnetic flux density decreases
• the field becomes weaker

This explains why a magnet usually attracts a magnetic object much more strongly when the object is close to one of its poles.

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Magnetic Flux Density Is Not the Same as Magnetic Flux

The terms magnetic flux density and magnetic flux sound similar but describe different quantities.

Magnetic flux density

Symbol:

B

Unit:

tesla (T)

It describes the magnetic field at a location.

Magnetic flux

Symbol:

Φ

Unit:

weber (Wb)

It describes the magnetic field passing through a particular area.

For a uniform field:

Φ = BA cos θ

where θ is the angle between the magnetic field and the normal to the surface.

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Understanding the Area Angle

The angle in:

Φ = BA cos θ

is measured between:

• the magnetic field B
• a line perpendicular to the surface, called the normal

If the field passes directly through the surface:

θ = 0°

cos 0° = 1

Therefore:

Φ = BA

If the field runs parallel to the surface:

θ = 90°

cos 90° = 0

Therefore:

Φ = 0

This distinction becomes especially important when studying electromagnetic induction.

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Tesla and Weber

From:

Φ = BA

we can rearrange:

B = Φ/A

Therefore:

1 T = 1 Wb/m²

So a magnetic flux density of 1 tesla can be interpreted as:

1 weber of magnetic flux per square metre

when the field passes perpendicular to the area.

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Magnetic Flux Density as a Vector

Because B is a vector, two magnetic fields with the same numerical strength may still be different if they point in different directions.

For example:

B₁ = 0.40 T east

and:

B₂ = 0.40 T west

have equal magnitudes but opposite directions.

When magnetic fields overlap, their vector directions must be considered.

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Direction of Magnetic Flux Density

At any point, the direction of B is:

tangent to the magnetic field line

For a bar magnet, outside the magnet:

N → S

Therefore, the magnetic flux density vector points along the field lines from north toward south outside the magnet.

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Uniform Magnetic Flux Density

A uniform magnetic field has the same magnetic flux density throughout a region.

This means B has approximately constant:

• magnitude
• direction

A uniform magnetic field is represented by:

straight, parallel, equally spaced field lines

For example:

→ → → → →

→ → → → →

→ → → → →

The constant spacing represents constant field strength.

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Non-Uniform Magnetic Flux Density

A non-uniform magnetic field has a magnetic flux density that changes with position.

The field around a bar magnet is an example.

Near the poles:

large B

Far from the magnet:

smaller B

The direction of B also changes as the field lines curve.

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Field Between the Poles of a Horseshoe Magnet

A horseshoe magnet places its north and south poles close together.

In the central region between the poles, the field lines may be:

• close together
• nearly parallel
• approximately equally spaced

This represents a relatively:

strong and approximately uniform magnetic field

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Factors Affecting the Field of a Permanent Magnet

The magnetic flux density produced by a permanent magnet depends on several factors.

These include:

• type of magnetic material
• strength of magnetization
• size and shape of the magnet
• distance from the magnet
• location relative to the poles
• presence of other magnetic materials

A strongly magnetized material can produce a larger B than a weakly magnetized one.

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Distance from a Magnet

Magnetic flux density generally decreases as distance from a magnet increases.

Therefore:

closer to magnet → larger B

farther from magnet → smaller B

The exact relationship depends on the shape of the magnet and where the measurement is made.

It should not automatically be assumed to follow a simple inverse-square relationship.

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Magnetic Material

Different magnetic materials can produce very different magnetic field strengths.

Strong permanent magnets are often made from materials such as:

• neodymium-iron-boron
• samarium-cobalt
• ferrite
• alnico

Their composition and internal magnetic structure affect the magnetic flux density they can produce.

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Magnetic Flux Density Around a Current-Carrying Wire

Electric current also produces a magnetic field.

Around a long straight wire, magnetic field lines form concentric circles.

For a long straight wire:

B = μ₀I/(2πr)

where:

• B = magnetic flux density
• μ₀ = permeability of free space
• I = current
• r = distance from the wire

This equation shows two important relationships.

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Effect of Current

From:

B = μ₀I/(2πr)

we see:

B ∝ I

Therefore:

greater current → stronger magnetic field

If current doubles while distance remains constant:

B doubles

This is an important way of controlling the strength of electromagnets.

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Effect of Distance from a Wire

The same equation shows:

B ∝ 1/r

Therefore:

greater distance → weaker magnetic field

For example, if the distance from a long straight wire doubles while current remains constant:

B becomes half as large

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Example: Current and Field Strength

Suppose a wire produces:

B = 2.0 mT

at a particular distance when:

I = 3.0 A

If the current increases to:

6.0 A

while the distance remains unchanged:

current has doubled.

Since:

B ∝ I

the magnetic flux density also doubles.

Therefore:

B = 4.0 mT

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Example: Distance from a Wire

Suppose:

B = 8.0 mT

at:

r = 2.0 cm

If the distance increases to:

r = 4.0 cm

the distance has doubled.

Since:

B ∝ 1/r

the field becomes half as strong.

Therefore:

B = 4.0 mT

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Magnetic Field of a Solenoid

A solenoid is a long coil of wire carrying electric current.

Inside a sufficiently long solenoid, the magnetic field can be approximately uniform.

For an ideal long solenoid:

B = μ₀nI

where:

• B = magnetic flux density
• μ₀ = permeability of free space
• n = number of turns per metre
• I = current

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Increasing the Field of a Solenoid

From:

B = μ₀nI

the magnetic field can be strengthened by:

• increasing current
• increasing the number of turns per unit length
• using a suitable magnetic core

Therefore:

larger I → larger B

and:

larger n → larger B

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Iron Cores

Placing a ferromagnetic core such as soft iron inside a coil can greatly strengthen the magnetic field.

The coil's magnetic field encourages magnetic domains within the iron to align.

The field from the magnetized core reinforces the field produced by the coil.

This principle is used in:

• electromagnets
• transformers
• relays
• motors
• lifting magnets

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Force on a Current-Carrying Wire

One important way to define or measure magnetic flux density involves the force on a current-carrying conductor.

For a straight conductor in a uniform magnetic field:

F = BIL sin θ

where:

• F = magnetic force (N)
• B = magnetic flux density (T)
• I = current (A)
• L = length of conductor in the field (m)
• θ = angle between current and magnetic field

If the conductor is perpendicular to the field:

θ = 90°

and:

sin 90° = 1

Therefore:

F = BIL

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Rearranging for Magnetic Flux Density

From:

F = BIL

we can calculate:

B = F/(IL)

when the wire is perpendicular to the field.

This provides another interpretation of the tesla.

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Meaning of One Tesla

Suppose:

I = 1 A

L = 1 m

and the conductor is perpendicular to the magnetic field.

If the magnetic force is:

F = 1 N

then:

B = F/(IL)

B = 1/(1 × 1)

B = 1 T

Therefore, a magnetic flux density of 1 T produces a force of 1 N on a 1 m conductor carrying 1 A when the conductor is perpendicular to the field.

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Example: Calculating B from Force

A wire carries:

I = 4.0 A

A length:

L = 0.20 m

is perpendicular to a magnetic field.

The wire experiences a force:

F = 0.12 N

Use:

B = F/(IL)

B = 0.12/(4.0 × 0.20)

B = 0.12/0.80

B = 0.15 T

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Example: Calculating Force

A wire of length:

L = 0.50 m

carries:

I = 3.0 A

perpendicular to a field:

B = 0.20 T

Use:

F = BIL

F = (0.20)(3.0)(0.50)

F = 0.30 N

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Effect of Angle

The full equation is:

F = BIL sin θ

The force is greatest when:

θ = 90°

because:

sin 90° = 1

The force is zero when:

θ = 0°

because:

sin 0° = 0

Therefore:

current perpendicular to field → maximum force

current parallel to field → zero magnetic force

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Magnetic Force on a Moving Charge

A moving electric charge can also experience a magnetic force.

The magnitude is:

F = Bqv sin θ

where:

• F = magnetic force
• B = magnetic flux density
• q = magnitude of charge
• v = speed of the charge
• θ = angle between velocity and magnetic field

This equation provides another way to understand what magnetic flux density represents.

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Example: Moving Charge

A particle has:

q = 2.0 × 10⁻⁶ C

and moves at:

v = 500 m/s

perpendicular to a magnetic field:

B = 0.40 T

Use:

F = Bqv

F = (0.40)(2.0 × 10⁻⁶)(500)

F = 4.0 × 10⁻⁴ N

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Magnetic Fields Do No Work on Moving Charges

The magnetic force on a moving charged particle is perpendicular to its velocity.

Therefore, a magnetic field can change the:

direction

of a charged particle's motion without directly changing its:

speed

or kinetic energy.

This is an important difference from electric fields, which can do work on charges and change their kinetic energy.

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Interpreting Field Diagrams

Suppose a field diagram shows three locations:

• A: field lines very close together
• B: field lines moderately spaced
• C: field lines widely separated

Then:

B_A > B_B > B_C

Point A has the greatest magnetic flux density.

Point C has the smallest.

No numerical calculation is necessary unless numerical field values are provided.

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Field Into and Out of the Page

Magnetic fields are three-dimensional, so diagrams need a way to represent fields perpendicular to the page.

A field pointing out of the page is represented by:

⊙

This can be imagined as seeing the point of an arrow coming toward you.

A field pointing into the page is represented by:

⊗

This can be imagined as seeing the feathers or crossed tail of an arrow moving away from you.

Therefore:

⊙ = out of page

⊗ = into page

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Uniform Field Diagrams

A uniform field pointing into the page might appear as:

⊗ ⊗ ⊗ ⊗

⊗ ⊗ ⊗ ⊗

⊗ ⊗ ⊗ ⊗

Equal spacing indicates approximately constant B.

A uniform field pointing out of the page might appear as:

⊙ ⊙ ⊙ ⊙

⊙ ⊙ ⊙ ⊙

⊙ ⊙ ⊙ ⊙

These symbols are extremely common in magnetism problems.

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Measuring Magnetic Flux Density

Magnetic flux density can be measured using instruments such as:

• Hall probes
• gaussmeters
• teslameters
• magnetic field sensors

A Hall probe can be moved through a magnetic field to investigate how B changes with position.

For example, students can measure B:

• near a magnet
• farther from the magnet
• between magnetic poles
• inside a solenoid

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Magnetic Flux Density and Magnetic Flux

Consider a uniform magnetic field:

B = 0.50 T

passing perpendicular to an area:

A = 0.20 m²

Use:

Φ = BA

Φ = (0.50)(0.20)

Φ = 0.10 Wb

This means 0.10 weber of magnetic flux passes through the area.

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Example: Finding Flux Density from Flux

Suppose:

Φ = 0.060 Wb

passes perpendicularly through:

A = 0.020 m²

Use:

B = Φ/A

B = 0.060/0.020

B = 3.0 T

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Effect of Area

If B remains constant:

Φ = BA

Increasing the area increases the magnetic flux passing through the surface.

However, this does not mean that the magnetic flux density has increased.

The magnetic field can have the same B while a larger area intercepts more total magnetic flux.

This distinction is important.

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Flux Density vs Flux Example

Imagine rain falling uniformly.

Magnetic flux density is somewhat analogous to:

how intensely rain is falling per unit area

Magnetic flux is somewhat analogous to:

the total amount of rain collected over an area

A larger container can collect more rain even though the rainfall intensity has not changed.

Similarly, a larger surface can have more magnetic flux through it even when B remains constant.

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Comparing Magnetic Fields

When comparing magnetic fields, consider:

Field-line density

Closer lines usually represent larger B.

Distance from source

Fields usually weaken with increasing distance.

Current

Larger currents generally produce stronger magnetic fields.

Number of coil turns

More turns per unit length can strengthen a solenoid's field.

Magnetic core

A suitable ferromagnetic core can greatly increase B.

Magnet material

Different permanent-magnet materials can produce different field strengths.

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Simple Problem-Solving Strategy

When solving magnetic flux density problems:

Identify the known quantities

For example:

F = 0.50 N

I = 5.0 A

L = 0.25 m

Select the correct equation

If the wire is perpendicular to the field:

B = F/(IL)

Substitute

B = 0.50/(5.0 × 0.25)

Calculate

B = 0.50/1.25

B = 0.40 T

Include the correct unit

Magnetic flux density must be expressed in:

tesla (T)

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Multi-Step Example

A straight wire carries:

I = 2.5 A

A length:

L = 0.40 m

lies perpendicular to a magnetic field.

The magnetic force is:

F = 0.30 N

Find the magnetic flux density

B = F/(IL)

B = 0.30/(2.5 × 0.40)

B = 0.30/1.00

B = 0.30 T

Now suppose the current doubles to:

5.0 A

while B and L remain unchanged.

Find the new force

F = BIL

F = (0.30)(5.0)(0.40)

F = 0.60 N

Doubling the current doubles the force when all other factors remain constant.

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Choosing the Correct Equation

Use:

B = F/(IL)

when finding magnetic flux density from the force on a perpendicular current-carrying conductor.

Use:

F = BIL

when the current is perpendicular to the field.

Use:

F = BIL sin θ

when the conductor is at an angle to the field.

Use:

B = μ₀I/(2πr)

for the field around an ideal long straight current-carrying wire.

Use:

B = μ₀nI

for an ideal long air-core solenoid.

Use:

Φ = BA

when a uniform field passes perpendicular to an area.

Use:

Φ = BA cos θ

when the surface is at another orientation, with θ measured from the surface normal.

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Common Mistakes

Mistake: Saying magnetic flux density is measured in webers.
Magnetic flux density is measured in teslas. Magnetic flux is measured in webers.

Mistake: Confusing B and Φ.
B describes the magnetic field at a location; Φ describes magnetic flux through an area.

Mistake: Saying closely spaced field lines represent a weak field.
They represent a stronger field.

Mistake: Assuming field-line density provides an exact numerical B value.
Ordinary diagrams generally provide qualitative comparisons unless a scale or numerical information is supplied.

Mistake: Forgetting to convert mT to T.
1 mT = 10⁻³ T.

Mistake: Forgetting the angle in F = BIL sin θ.
The simplified equation F = BIL applies when the conductor is perpendicular to the field.

Mistake: Assuming increasing area increases B.
Increasing area can increase magnetic flux without changing magnetic flux density.

Mistake: Confusing ⊙ and ⊗.
⊙ means out of the page; ⊗ means into the page.

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Key Terms

Magnetic flux density — vector quantity describing the strength and direction of a magnetic field at a point.

B — symbol used for magnetic flux density.

Tesla (T) — SI unit of magnetic flux density.

Magnetic field — region where magnetic forces can act.

Magnetic field line — representation showing magnetic-field direction and relative strength.

Field-line density — concentration of field lines used to represent relative magnetic field strength.

Magnetic flux — measure of magnetic field passing through an area.

Weber (Wb) — SI unit of magnetic flux.

Uniform magnetic field — field with approximately constant magnitude and direction.

Solenoid — coil of wire that produces a magnetic field when carrying current.

Hall probe — sensor used to measure magnetic flux density.

Ferromagnetic core — magnetic material placed within a coil to strengthen its magnetic field.

Permeability — property describing how a material or medium responds to a magnetic field.

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Key Takeaways

• Magnetic flux density describes the strength and direction of a magnetic field.
• Magnetic flux density is represented by B.
• B is a vector quantity.
• The SI unit of B is the tesla (T).
• 1 mT = 10⁻³ T and 1 μT = 10⁻⁶ T.
• Larger B means a stronger magnetic field.
• Closely spaced field lines represent greater magnetic flux density.
• Magnetic fields around permanent magnets generally weaken with increasing distance.
• Around a long straight wire, increasing current increases B.
• Around a long straight wire, increasing distance decreases B.
• Increasing current strengthens the field of a solenoid.
• Increasing turns per unit length strengthens a solenoid's field.
• A ferromagnetic core can greatly increase the field of an electromagnet.
• Magnetic flux density and magnetic flux are different quantities.
• Magnetic flux density is measured in teslas.
• Magnetic flux is measured in webers.
• For a perpendicular current-carrying conductor, F = BIL.
• For a uniform field perpendicular to an area, Φ = BA.
• ⊙ represents a field out of the page.
• ⊗ represents a field into the page.

Check Your Understanding

1. Define magnetic flux density.
2. State the symbol used for magnetic flux density.
3. State the SI unit of magnetic flux density.
4. Convert 250 mT into teslas.
5. Convert 0.003 T into milliteslas.
6. What does closely spaced magnetic field lines indicate?
7. Where is B generally greatest around a bar magnet?
8. Explain how distance from a magnet affects magnetic flux density.
9. How does increasing current affect the magnetic field around a straight wire?
10. How does increasing distance from a straight current-carrying wire affect B?
11. Give three ways of increasing the magnetic field inside a solenoid.
12. Explain the difference between magnetic flux and magnetic flux density.
13. What does the symbol ⊙ represent?
14. What does the symbol ⊗ represent?
15. A 0.50 m wire carries 4.0 A perpendicular to a 0.30 T field. Calculate the magnetic force.
16. A 0.20 m wire carrying 5.0 A experiences a force of 0.40 N perpendicular to a magnetic field. Calculate B.
17. A uniform magnetic field of 0.60 T passes perpendicularly through an area of 0.25 m². Calculate the magnetic flux.
18. A magnetic flux of 0.080 Wb passes perpendicularly through an area of 0.040 m². Calculate B.
19. Explain why increasing the area through which a field passes can increase magnetic flux without increasing magnetic flux density.
20. Describe how a Hall probe could be used to investigate how magnetic flux density changes with distance from a permanent magnet.