Electromagnetic Force

How Magnetic Fields and Currents Interact

When an electric current flows through a conductor, it produces a magnetic field around the conductor. If that conductor is placed in another magnetic field, the two magnetic fields interact.

This interaction can produce a force on the conductor.

This is called the motor effect and is one of the most important applications of electromagnetism:

current + magnetic field → force

For a straight current-carrying conductor:

F = BIL sin θ

where:

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

---

The Motor Effect

A current-carrying wire produces its own magnetic field.

When the wire is placed inside an external magnetic field, the fields interact.

On one side of the wire, the fields may reinforce each other.

On the other side, they may partially oppose each other.

The resulting field pattern is asymmetric and the conductor experiences a net force.

This is the basis of the:

motor effect

---

Force on the Moving Charges

Electric current consists of moving charge.

Individual moving charges in a magnetic field experience:

F = |q|vB sin θ

where:

• |q| = charge magnitude
• v = charge velocity
• B = magnetic flux density
• θ = angle between v and B

The magnetic forces acting on the moving charges are transferred to the conductor.

At the macroscopic level, this gives:

F = BIL sin θ

for a straight conductor.

---

Maximum Electromagnetic Force

If the conductor is perpendicular to the magnetic field:

θ = 90°

Since:

sin 90° = 1

the equation becomes:

F = BIL

The force is maximum.

Therefore:

current perpendicular to field → maximum force

---

Zero Electromagnetic Force

If the conductor is parallel to the magnetic field:

θ = 0°

Since:

sin 0° = 0

then:

F = 0

Therefore:

current parallel to field → no magnetic force

This angular dependence is important when analyzing motors and other electromagnetic devices.

---

Effect of Angle

In general:

F = BIL sin θ

Therefore:

The force depends on the component of the magnetic field perpendicular to the current.

---

Direction of the Force

The electromagnetic force acts perpendicular to both:

• current direction
• magnetic-field direction

For conventional current, the direction can be determined using:

Fleming's Left-Hand Rule

---

Fleming's Left-Hand Rule

Hold your left:

• thumb
• first finger
• second finger

at right angles to one another.

First finger

Points in the direction of the magnetic:

Field

Second finger

Points in the direction of conventional:

Current

Thumb

Points in the direction of:

Force or Motion

A useful memory aid is:

First = Field

seCond = Current

thuMb = Motion

---

Conventional Current

Fleming's Left-Hand Rule uses:

conventional current

Conventional current is defined as the direction positive charge would move.

In a metal conductor, electrons actually drift in the opposite direction.

Unless specifically told otherwise, use:

conventional current direction

when applying the rule.

---

Reversing the Current

Suppose a conductor experiences a force upward.

If the current is reversed while the magnetic field remains unchanged:

force direction reverses

Therefore:

upward → downward

This is important in devices where the direction of motion must be controlled.

---

Reversing the Magnetic Field

If the magnetic-field direction is reversed while current remains unchanged:

force direction reverses

Again:

upward → downward

---

Reversing Both

If both:

• current direction
• magnetic-field direction

are reversed simultaneously, the force direction remains unchanged.

This is because each reversal individually reverses the force, producing two reversals overall.

---

Factors Affecting Electromagnetic Force

From:

F = BIL sin θ

four main factors affect force.

Magnetic flux density

B ↑ → F ↑

A stronger magnetic field produces a larger force.

Current

I ↑ → F ↑

A larger current produces a larger force.

Length in the field

L ↑ → F ↑

A longer section of conductor inside the field produces a larger force.

Angle

Force depends on:

sin θ

Maximum at 90°.

Zero at 0° or 180°.

---

Example: Calculating Force

A wire carries:

I = 4.0 A

through a magnetic field:

B = 0.50 T

The length of wire inside the field is:

L = 0.30 m

The wire is perpendicular to the field.

Use:

F = BIL

F = (0.50)(4.0)(0.30)

F = 0.60 N

---

Example: Force at an Angle

A conductor has:

B = 0.80 T

I = 3.0 A

L = 0.40 m

θ = 30°

Use:

F = BIL sin θ

F = (0.80)(3.0)(0.40)(sin 30°)

F = (0.80)(3.0)(0.40)(0.50)

F = 0.48 N

---

Example: Finding Current

A wire experiences a force:

F = 1.2 N

in a field:

B = 0.50 T

The length in the field is:

L = 0.80 m

The conductor is perpendicular to the field.

Use:

F = BIL

Rearrange:

I = F/(BL)

I = 1.2/[(0.50)(0.80)]

I = 1.2/0.40

I = 3.0 A

---

Example: Finding Magnetic Flux Density

A conductor experiences:

F = 0.36 N

when carrying:

I = 2.0 A

The length inside the field is:

L = 0.30 m

The wire is perpendicular to the field.

Use:

B = F/(IL)

B = 0.36/[(2.0)(0.30)]

B = 0.60 T

---

Example: Unit Conversion

A 12 cm wire carries 5.0 A perpendicular to a 0.40 T magnetic field.

First convert:

12 cm = 0.12 m

Then:

F = BIL

F = (0.40)(5.0)(0.12)

F = 0.24 N

Always convert length to metres before using the equation.

---

Forces on Parallel Current-Carrying Wires

Two current-carrying wires can exert forces on each other.

Each wire produces a magnetic field.

The other wire lies inside that field and therefore experiences a magnetic force.

For parallel wires:

currents in the same direction → attraction

currents in opposite directions → repulsion

---

Why Parallel Currents Attract

Consider two parallel wires carrying current in the same direction.

Wire A produces a magnetic field at Wire B.

Wire B carries current inside this field and experiences a force toward Wire A.

Similarly, Wire B produces a field that causes Wire A to experience a force toward Wire B.

Therefore:

the wires attract

---

Why Opposite Currents Repel

If the currents flow in opposite directions, the magnetic force directions reverse.

Each wire is pushed away from the other.

Therefore:

opposite parallel currents repel

This is a direct consequence of current-produced magnetic fields interacting with other currents.

---

Force Between Long Parallel Wires

For two long parallel wires separated by distance r:

F/L = μ₀I₁I₂/(2πr)

where:

• F/L = force per unit length (N/m)
• μ₀ = permeability of free space
• I₁ = current in first wire
• I₂ = current in second wire
• r = separation between wires

This equation shows:

larger currents → larger force

greater separation → smaller force

---

Current-Carrying Coils

A coil placed in a magnetic field can experience forces on different sections.

Opposite sides of the coil carry current in opposite directions.

Therefore, the forces on those sides also point in opposite directions.

Rather than simply moving the entire coil sideways, these forces can create:

torque

or a turning effect.

---

Torque on a Coil

For an ideal coil:

τ = NBIA sin θ

where:

• τ = torque (N·m)
• N = number of turns
• B = magnetic flux density (T)
• I = current (A)
• A = coil area (m²)
• θ = angle between the coil's magnetic moment and B

This equation shows that torque can be increased by increasing:

• number of turns
• magnetic-field strength
• current
• coil area

---

Electromagnetic Force in Motors

An electric motor converts:

electrical energy → mechanical energy

The basic process is:

current flows through coil

↓

coil lies in magnetic field

↓

opposite sides experience opposite forces

↓

forces create torque

↓

rotor turns

---

Why a Motor Rotates

Consider a rectangular coil between magnetic poles.

On one side of the coil, current may flow into the page.

On the opposite side, current flows out of the page.

Using Fleming's Left-Hand Rule:

• one side experiences an upward force
• the other experiences a downward force

These forces form a couple.

The resulting torque rotates the coil.

---

The Split-Ring Commutator

In a simple DC motor, the current must reverse approximately every half-turn so that the torque continues in the same overall rotational direction.

The:

split-ring commutator

performs this function.

It reverses current through the coil as the motor rotates.

Therefore, the magnetic forces reverse at the correct time and continuous rotation can occur.

---

Increasing Motor Force and Torque

A motor can produce greater electromagnetic force by increasing:

• B
• I
• effective conductor length

Motor torque can also be increased by:

• increasing number of coil turns
• increasing coil area
• optimizing coil orientation

Engineering involves balancing these improvements against:

• heating
• size
• mass
• electrical power
• material limitations

---

Electromagnetic Force in Loudspeakers

A loudspeaker also uses the motor effect.

A voice coil sits inside the field of a permanent magnet.

An audio current passes through the coil.

The coil experiences an electromagnetic force.

---

How a Loudspeaker Produces Sound

The audio current continually changes.

When current flows one way:

force acts one way

When current reverses:

force reverses

Therefore, the voice coil moves back and forth.

The voice coil is attached to the speaker cone.

So:

electrical signal

↓

changing current

↓

changing electromagnetic force

↓

voice coil vibrates

↓

cone vibrates

↓

air pressure changes

↓

sound wave

---

Force and Speaker Volume

For the simplified relationship:

F = BIL

increasing current increases force.

A larger varying current can therefore produce larger cone movements, within the speaker's operating limits.

Larger vibration amplitude generally produces a sound wave with greater amplitude.

This is associated with increased:

sound intensity and perceived loudness

---

Electromagnetic Relays

A relay uses electromagnetic forces differently.

Current flows through a coil.

The coil produces a magnetic field.

The field magnetizes or attracts a ferromagnetic armature.

The armature moves and changes the position of electrical contacts.

Therefore:

electrical signal → magnetic field → mechanical movement → switching

---

Solenoid Actuators

A solenoid actuator converts electrical energy into approximately linear motion.

When current flows through a coil:

• a magnetic field is produced
• a ferromagnetic plunger experiences a force
• the plunger moves

These devices are used in:

• valves
• door locks
• vehicle systems
• vending machines
• industrial machines
• robotic systems

---

Moving-Coil Meters

A moving-coil meter places a current-carrying coil in a magnetic field.

The electromagnetic forces produce torque.

Greater current produces greater torque.

The coil rotates against a restoring mechanism.

A pointer attached to the coil can therefore indicate the current or another electrical quantity.

This principle is used in traditional:

• galvanometers
• ammeters
• voltmeters

---

Electromagnetic Force in Particle Beams

The same magnetic-force principle applies to moving individual charges.

For a particle:

F = |q|vB sin θ

A magnetic field can therefore:

• bend particle beams
• guide charged particles
• produce circular motion
• help confine plasma

This is used in:

• particle accelerators
• mass spectrometers
• fusion research
• particle detectors

---

Macroscopic and Microscopic Views

There are two useful ways to understand electromagnetic force.

Microscopic view

Moving individual charges experience:

F = |q|vB sin θ

Macroscopic view

A current-carrying conductor experiences:

F = BIL sin θ

The macroscopic motor effect ultimately results from magnetic forces acting on moving charges inside the conductor.

---

Magnetic Force Does Not Create Energy

A magnetic field can exert forces and redirect moving charges, but it does not create energy.

In motors, the energy ultimately comes from the:

electrical power supply

The magnetic field provides a mechanism that allows electrical energy to be converted into mechanical motion.

Therefore:

magnetic interaction enables energy conversion

rather than generating energy from nothing.

---

Power in an Electromechanical Device

Electrical input power can be calculated using:

P = VI

A motor converts part of this electrical power into useful mechanical power.

However, some energy is dissipated through:

• resistance heating
• friction
• sound
• magnetic losses

Therefore:

P_input = P_useful + P_losses

---

Efficiency

Motor efficiency can be calculated using:

efficiency = useful output power / input power

or:

efficiency (%) = (useful output power / input power) × 100

For example, if a motor receives:

P_input = 500 W

and produces:

P_useful = 400 W

then:

efficiency = 400/500 × 100

efficiency = 80%

The remaining 20% is transferred into less useful forms such as heat and sound.

---

Example: Motor Force

A motor conductor has:

B = 0.75 T

I = 6.0 A

L = 0.20 m

The conductor is perpendicular to the field.

F = BIL

F = (0.75)(6.0)(0.20)

F = 0.90 N

If the motor contains many conductor sections experiencing similar forces, their combined effects can produce substantial torque.

---

Example: Effect of Doubling Current

A conductor initially experiences:

F = 0.40 N

The current is doubled while B, L and θ remain unchanged.

Since:

F ∝ I

the new force is:

F = 0.80 N

---

Example: Effect of Doubling Field Strength

A wire experiences:

F = 0.25 N

The magnetic flux density doubles.

Since:

F ∝ B

the new force is:

F = 0.50 N

---

Example: Changing Several Variables

A conductor initially experiences force F.

The current doubles:

I → 2I

The magnetic flux density triples:

B → 3B

The length and angle remain unchanged.

Since:

F ∝ BI

the new force is:

(2)(3)F

Therefore:

F_new = 6F

---

Example: Force at 60°

A wire has:

B = 0.40 T

I = 5.0 A

L = 0.50 m

θ = 60°

Use:

F = BIL sin θ

F = (0.40)(5.0)(0.50)(sin 60°)

F = (1.0)(0.866)

F ≈ 0.87 N

If the conductor were perpendicular to the field, the force would be:

F = (0.40)(5.0)(0.50)

F = 1.0 N

The angled conductor therefore experiences a smaller force.

---

Example: Multi-Step Electromagnetic Problem

A 25 cm conductor carries 8.0 A in a 0.60 T magnetic field at 30°.

Step 1: Convert length

25 cm = 0.25 m

Step 2: Select equation

F = BIL sin θ

Step 3: Substitute

F = (0.60)(8.0)(0.25)(sin 30°)

F = (0.60)(8.0)(0.25)(0.50)

F = 0.60 N

If the current is reversed, the magnitude remains:

0.60 N

but the force direction reverses.

---

Analyzing Electromagnetic Devices

When analyzing a device that uses electromagnetic force, ask:

Where does the current flow?

Identify the current-carrying conductor or coil.

Where is the magnetic field?

Identify the permanent magnet, electromagnet or other field source.

What is the angle between I and B?

This determines the force magnitude.

What is the force direction?

Use Fleming's Left-Hand Rule.

What does the force do?

It may produce:

• rotation
• vibration
• linear movement
• switching
• beam deflection

Where does the energy come from?

Usually from an electrical power source.

---

Comparing Devices

The applications differ, but the underlying electromagnetic principles are closely related.

---

Common Mistakes

Mistake: Saying the force is always F = BIL.
This simplified equation applies when current is perpendicular to the magnetic field. In general, use F = BIL sin θ.

Mistake: Using centimetres directly for L.
Convert length to metres.

Mistake: Saying current parallel to the magnetic field experiences maximum force.
Parallel current produces zero magnetic force.

Mistake: Using the right-hand grip rule to find motor force direction.
For conventional current in motor-effect problems, use Fleming's Left-Hand Rule.

Mistake: Forgetting that reversing current reverses the force.

Mistake: Forgetting that reversing the magnetic field also reverses the force.

Mistake: Saying parallel currents in the same direction repel.
They attract.

Mistake: Saying opposite parallel currents attract.
They repel.

Mistake: Saying a magnetic field supplies the energy used by a motor.
The motor's energy comes from its electrical supply.

Mistake: Confusing force and torque.
Forces acting at suitable positions can produce torque.

---

Key Terms

Electromagnetic force — force associated with interactions between electric charges, currents, electric fields and magnetic fields.

Motor effect — force experienced by a current-carrying conductor placed in a magnetic field.

Magnetic flux density (B) — measure of magnetic-field strength and direction, measured in teslas.

Current (I) — rate of flow of electric charge.

Conductor — material through which electric charge can move.

Fleming's Left-Hand Rule — rule used to determine force direction on a current-carrying conductor in a magnetic field.

Conventional current — defined direction of positive-charge flow.

Torque — turning effect produced by a force.

Couple — pair of forces producing a turning effect.

Rotor — rotating part of an electric motor.

Voice coil — current-carrying coil in a loudspeaker that experiences electromagnetic force.

Solenoid actuator — device using electromagnetism to produce controlled linear movement.

Armature — movable magnetic component in an electromagnetic device.

Electromechanical device — device that converts between electrical and mechanical forms of energy or signals.

---

Key Takeaways

• Electric currents produce magnetic fields.
• A current-carrying conductor can experience a force in an external magnetic field.
• This interaction is called the motor effect.
• The general force equation is F = BIL sin θ.
• When the conductor is perpendicular to the field, F = BIL.
• Force is maximum when current and field are perpendicular.
• Force is zero when current and field are parallel.
• Increasing B increases electromagnetic force.
• Increasing I increases electromagnetic force.
• Increasing the conductor length within the field increases force.
• Fleming's Left-Hand Rule determines force direction for conventional current.
• Reversing current reverses force direction.
• Reversing the magnetic field reverses force direction.
• Reversing both current and field leaves the force direction unchanged.
• Parallel currents in the same direction attract.
• Parallel currents in opposite directions repel.
• Forces on a current-carrying coil can produce torque.
• Electric motors use electromagnetic force to produce rotation.
• Loudspeakers use electromagnetic force to produce vibration.
• Relays, solenoid actuators and moving-coil meters also use electromagnetic interactions.
• Electromagnetic forces provide an important mechanism for converting electrical energy into mechanical motion.

Check Your Understanding

1. Explain what is meant by the motor effect.
2. State the general equation for force on a straight current-carrying conductor in a magnetic field.
3. What does each quantity in F = BIL sin θ represent?
4. At what angle is electromagnetic force maximum?
5. At what angle is electromagnetic force zero?
6. State Fleming's Left-Hand Rule.
7. What happens to force direction if current is reversed?
8. What happens if the magnetic field is reversed?
9. What happens if both current and magnetic-field directions are reversed?
10. A 0.40 m wire carries 3.0 A perpendicular to a 0.50 T field. Calculate the force.
11. A 20 cm wire carries 6.0 A perpendicular to a 0.80 T field. Calculate the force.
12. A conductor carries 4.0 A through a 0.60 T field at 30°. Its length in the field is 0.50 m. Calculate the force.
13. Explain how increasing current affects electromagnetic force.
14. Explain how increasing magnetic flux density affects electromagnetic force.
15. Describe the force between two parallel wires carrying currents in the same direction.
16. Describe the force between two parallel wires carrying currents in opposite directions.
17. Explain how electromagnetic forces produce torque in an electric motor.
18. Explain how a loudspeaker uses electromagnetic force to produce sound.
19. Compare the use of electromagnetic force in a motor, loudspeaker and solenoid actuator.
20. A motor designer wants to increase the electromagnetic force without increasing the conductor length. Describe at least three changes that could be considered and explain the physics behind each.

Hall Effect and Applications

What Is the Hall Effect?

The Hall Effect occurs when a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current.

The moving charge carriers experience a magnetic force. They are pushed toward one side of the material, producing a separation of charge.

This creates a voltage across the material called the:

Hall voltage, V_H

The Hall Effect therefore provides a direct connection between:

moving charge + magnetic field → charge separation → measurable voltage

It is widely used to measure magnetic fields and detect position, motion, speed and electric current.

---

The Basic Hall Effect Setup

Consider a thin rectangular conducting or semiconducting strip.

A current flows along its length.

Now place the strip in a magnetic field perpendicular to the current.

For example:

• current → to the right
• magnetic field → into the page
• magnetic force → toward one side of the strip

Charge carriers accumulate along one side.

The opposite side is left with the opposite net charge.

A potential difference develops across the width.

This is the:

Hall voltage

---

Why Moving Charges Are Deflected

A moving charged particle in a magnetic field experiences:

F_B = |q|vB sin θ

where:

• F_B = magnetic force
• |q| = charge magnitude
• v = charge-carrier drift speed
• B = magnetic flux density
• θ = angle between v and B

If the motion is perpendicular to the magnetic field:

θ = 90°

so:

F_B = |q|vB

The force is perpendicular to both the motion and magnetic field.

This pushes charge carriers sideways.

---

Direction of Magnetic Force

For a positive charge, the magnetic-force direction can be found from:

F = q(v × B)

A right-hand rule can determine the direction.

For negative charge carriers such as electrons, the force direction is opposite to that for a positive charge moving in the same direction.

This is important because Hall measurements can provide information about the type of charge carriers present in a material.

---

Charge Separation

As charge carriers move sideways, they accumulate on one side of the material.

This creates:

• excess charge on one side
• opposite net charge on the other side

The separated charges create an internal electric field across the material.

This field is called the:

Hall field

---

The Hall Electric Field

As more charge accumulates, the Hall electric field becomes stronger.

The electric force on a charge is:

F_E = |q|E_H

where E_H is the Hall electric-field strength.

The electric force acts opposite to the sideways magnetic deflection.

Eventually:

electric force = magnetic force

At this point, charge carriers no longer continue accumulating significantly on one side.

---

Force Balance

At equilibrium:

F_E = F_B

Therefore:

|q|E_H = |q|v_dB

where v_d is the drift speed.

Cancel |q|:

E_H = v_dB

This relationship explains how the Hall electric field depends on:

• charge-carrier drift speed
• magnetic flux density

---

From Hall Field to Hall Voltage

If the width across which the Hall voltage is measured is w:

V_H = E_Hw

Since:

E_H = v_dB

then:

V_H = v_dBw

This shows that a stronger magnetic field produces a larger Hall voltage when other quantities remain constant.

---

Current and Charge-Carrier Density

Electric current in a conductor can be related to charge-carrier motion by:

I = nqAv_d

where:

• I = current (A)
• n = number of charge carriers per unit volume (m⁻³)
• q = charge per carrier (C)
• A = cross-sectional area (m²)
• v_d = drift speed (m/s)

For a Hall sample with width w and thickness t:

A = wt

Therefore:

I = nqwtv_d

Rearranging:

v_d = I/(nqwt)

---

Hall Voltage Equation

Starting with:

V_H = v_dBw

and substituting:

v_d = I/(nqwt)

gives:

V_H = [I/(nqwt)]Bw

The width cancels:

V_H = IB/(nqt)

For magnitude:

|V_H| = IB/(n|q|t)

where:

• V_H = Hall voltage (V)
• I = current (A)
• B = magnetic flux density (T)
• n = charge-carrier number density (m⁻³)
• q = charge per carrier (C)
• t = thickness of the Hall element (m)

This is an important Hall Effect relationship.

---

Factors Affecting Hall Voltage

From:

|V_H| = IB/(n|q|t)

we can identify several important relationships.

Current

V_H ∝ I

Larger current → larger Hall voltage.

Magnetic flux density

V_H ∝ B

Stronger magnetic field → larger Hall voltage.

Charge-carrier density

V_H ∝ 1/n

Lower carrier density → larger Hall voltage.

Thickness

V_H ∝ 1/t

Thinner Hall element → larger Hall voltage.

Charge per carrier

Larger |q| reduces Hall voltage for the same carrier density and other conditions.

---

Why Semiconductors Are Useful

Hall Effect devices are often made using semiconductors rather than ordinary metals.

Semiconductors can have much lower charge-carrier densities than metals.

Since:

V_H ∝ 1/n

a lower carrier density can produce a larger and easier-to-measure Hall voltage.

This makes semiconductors particularly useful for Hall sensors.

---

Example: Effect of Magnetic Field

A Hall sensor produces:

V_H = 20 mV

in a magnetic field:

B = 0.10 T

The field is increased to:

B = 0.30 T

Since:

V_H ∝ B

the field has tripled.

Therefore:

V_H = 3(20)

V_H = 60 mV

provided the other quantities remain constant.

---

Example: Effect of Current

A Hall element produces:

V_H = 5.0 mV

when:

I = 10 mA

The current is increased to:

20 mA.

Since:

V_H ∝ I

doubling current doubles the Hall voltage:

V_H = 10 mV

---

Example: Effect of Thickness

Two Hall elements are identical except for thickness.

Sensor A:

t = 1.0 mm

Sensor B:

t = 0.50 mm

Since:

V_H ∝ 1/t

Sensor B is half as thick.

Therefore, under the same conditions:

Sensor B produces twice the Hall voltage.

---

Example: Calculating Hall Voltage

Suppose:

I = 0.020 A

B = 0.50 T

n = 5.0 × 10²² m⁻³

|q| = 1.60 × 10⁻¹⁹ C

t = 1.0 × 10⁻³ m

Use:

V_H = IB/(n|q|t)

V_H = (0.020)(0.50) / [(5.0 × 10²²)(1.60 × 10⁻¹⁹)(1.0 × 10⁻³)]

The denominator is:

8.0

Therefore:

V_H = 0.010/8.0

V_H = 1.25 × 10⁻³ V

V_H = 1.25 mV

---

Hall Coefficient

Another useful quantity is the:

Hall coefficient, R_H

For a simple material with one dominant charge-carrier type:

R_H = 1/(nq)

The sign of R_H depends on the sign of the dominant charge carrier.

The Hall voltage can then be written:

V_H = R_HIB/t

The Hall coefficient can therefore provide information about:

• carrier concentration
• carrier type

---

Identifying Charge Carriers

One historically important use of the Hall Effect is determining whether the dominant charge carriers behave as:

• negative carriers
• positive carriers

The polarity of the Hall voltage depends on the charge-carrier sign.

In many metals, the dominant mobile carriers are electrons.

In semiconductors, conduction can be dominated by:

• electrons
• holes

Hall measurements are therefore valuable in semiconductor physics.

---

What Is a Hall Probe?

A Hall probe is a device that uses the Hall Effect to measure magnetic flux density.

A Hall probe contains a small Hall element carrying a known current.

When placed in a magnetic field:

magnetic field → Hall voltage

The instrument measures this voltage and converts it into a value of:

B

usually measured in:

• teslas (T)
• milliteslas (mT)
• microteslas (μT)

---

How a Hall Probe Measures Magnetic Fields

The process can be summarized:

1. A known current passes through the Hall element.
2. The probe is placed in the magnetic field.
3. Moving charge carriers experience magnetic force.
4. Charge separation occurs.
5. A Hall voltage develops.
6. The voltage is measured.
7. Calibration converts voltage into magnetic flux density.

Therefore:

Hall voltage → measurement of B

---

Probe Orientation

Hall voltage depends on the component of magnetic field perpendicular to the current-carrying Hall element.

If the relevant angle is θ:

V_H ∝ B sin θ

Maximum response occurs when:

θ = 90°

If the magnetic field is oriented so that the relevant perpendicular component is zero:

Hall voltage ≈ 0

Correct probe orientation is therefore essential when measuring magnetic fields.

---

Finding Magnetic-Field Direction

A Hall probe can also help determine magnetic-field direction.

Changing the direction of B reverses the direction of magnetic force on the charge carriers.

Therefore, the polarity of the Hall voltage reverses.

A properly designed instrument can use this to distinguish:

+B from −B

along its measurement axis.

---

Mapping Magnetic Fields

Hall probes can measure the magnetic field at many different positions.

For example, a student could move a Hall probe around:

• a bar magnet
• a solenoid
• an electromagnet

and record B at each location.

This allows the field to be mapped quantitatively.

Unlike iron filings, a Hall probe provides numerical measurements of magnetic flux density.

---

Example: Solenoid Investigation

Suppose a Hall probe is placed at the centre of a solenoid.

Current through the solenoid is gradually increased.

For an ideal long solenoid:

B = μ₀nI

Therefore, increasing I increases B.

The Hall probe detects this because:

B ↑ → V_H ↑

This experiment can be used to investigate the relationship between:

solenoid current and magnetic flux density

---

Hall Effect Sensors

A Hall Effect sensor uses the Hall Effect to detect a magnetic field.

Modern Hall sensors often contain:

• Hall element
• amplifier
• signal-processing electronics
• switching or digital circuitry

A very small Hall voltage can therefore be amplified and converted into a useful electrical output.

---

Analogue Hall Sensors

An analogue Hall sensor can produce an output that varies continuously with magnetic-field strength.

For example:

weak field → small output

stronger field → larger output

These sensors are useful when the system needs to measure:

how strong the magnetic field is

rather than simply detecting whether a magnet is present.

---

Digital Hall Sensors

A digital Hall sensor can operate like an electronic switch.

For example:

magnet far away → OFF

magnet approaches → ON

The Hall element detects the field, while internal electronics determine whether the field exceeds a particular threshold.

This is useful for:

• position detection
• limit switches
• door sensors
• speed measurement

---

Detecting Position

A magnet can be attached to a moving object.

A stationary Hall sensor detects when the magnet approaches.

Therefore, the sensor can determine whether an object has reached a particular position.

Applications include:

• robotic mechanisms
• industrial machinery
• vehicle systems
• electronic controls
• automatic doors

---

Detecting Rotational Speed

Hall sensors can measure rotational speed.

A small magnet is attached to a rotating wheel or shaft.

Each time the magnet passes the Hall sensor, the sensor produces a pulse.

If the sensor detects:

N pulses per second

and there is one magnet on the wheel:

rotation frequency = N rotations per second

If there are multiple magnets, the number of pulses per revolution must be considered.

---

Example: Rotational Speed

A wheel contains one magnet.

A Hall sensor detects:

20 pulses per second.

Therefore:

f = 20 revolutions/s

To convert to revolutions per minute:

rpm = 20 × 60

rpm = 1200

If the wheel had two equally spaced magnets, 20 pulses/s would correspond to only:

10 revolutions/s

or:

600 rpm

---

Measuring Wheel Speed

Hall sensors can be used to monitor rotating wheels.

If wheel radius is known, rotational information can be converted into linear speed.

For a wheel:

v = 2πrf

where:

• v = linear speed
• r = wheel radius
• f = rotation frequency

This principle is useful in:

• bicycles
• vehicles
• industrial machinery
• robotics

---

Brushless DC Motors

Hall Effect sensors are commonly used in some brushless DC motors.

The sensor detects the position of the rotor magnets.

The motor controller uses this information to determine when to switch current through different coils.

The sequence is:

rotor moves

↓

Hall sensor detects magnetic position

↓

controller determines rotor position

↓

current switches to appropriate winding

↓

electromagnetic torque continues

Hall sensors therefore help replace the mechanical commutator used in traditional brushed DC motors.

---

Current Sensors

Hall Effect sensors can also measure electric current without requiring the sensor electronics to be directly in the main current path.

A current-carrying wire produces a magnetic field.

For a long straight wire:

B = μ₀I/(2πr)

Therefore:

current → magnetic field → Hall voltage

A Hall sensor measures the magnetic field and the system determines the current.

---

Advantages of Hall Current Sensors

Hall-based current measurement can provide:

• electrical isolation
• measurement of large currents
• low interference with the main circuit
• measurement of DC as well as changing currents
• improved safety in high-power systems

These sensors are common in:

• electric vehicles
• battery systems
• power supplies
• industrial electronics
• renewable-energy systems

---

Contactless Switching

Because Hall sensors respond to magnetic fields, they do not require direct mechanical contact with the moving object.

This allows:

contactless sensing

Advantages can include:

• reduced mechanical wear
• long operating life
• sealed operation
• high reliability
• operation in dusty or dirty environments

---

Door and Lid Sensors

A magnet can be mounted on a door or lid and a Hall sensor on the frame.

Door closed:

magnet near sensor

Door open:

magnet moves away

The electronic system detects the change.

This can be used in:

• appliances
• laptops
• security systems
• industrial equipment

---

Smartphones and Portable Electronics

Hall sensors can be used in portable electronics to detect nearby magnets.

For example, a device may detect:

• magnetic covers
• accessories
• opening or closing mechanisms

The sensor can then trigger a software or hardware response.

---

Automotive Applications

Vehicles use Hall Effect sensors extensively.

Applications can include detecting:

• wheel rotation
• crankshaft position
• camshaft position
• motor position
• pedal position
• steering position
• current in electrical systems

Because Hall sensors can operate without direct mechanical contact, they are well suited to many vehicle systems.

---

Robotics

Robots often need information about the position of moving components.

Hall sensors can detect magnets attached to:

• joints
• wheels
• motors
• grippers
• linear mechanisms

The resulting electrical signal can be sent to a controller.

The controller can then adjust motor operation.

This creates a:

feedback system

---

Industrial Automation

Hall sensors can monitor:

• machine position
• shaft speed
• motor operation
• current
• magnetic-field strength

Because the sensing can be contactless, Hall sensors are useful in automated machinery where mechanical switches might wear out.

---

Hall Effect vs Mechanical Switch

---

Hall Effect vs Reed Switch

A reed switch can also respond to a magnetic field.

However, a traditional reed switch contains mechanical contacts that physically move.

A Hall sensor is electronic and has no comparable mechanical switching contacts.

Hall sensors are therefore useful when:

• rapid switching
• long lifetime
• electronic integration
• precise sensing

are important.

---

Measuring Magnetic Field Strength

Suppose a Hall probe has been calibrated so that:

10 mV corresponds to 0.20 T.

If the relationship is linear, then:

20 mV corresponds to:

0.40 T

and:

5 mV corresponds to:

0.10 T

This illustrates how a Hall probe converts a small voltage into a magnetic-field measurement.

---

Example: Finding B from Hall Voltage

Suppose:

V_H = 2.0 × 10⁻³ V

I = 0.040 A

n = 2.5 × 10²² m⁻³

|q| = 1.60 × 10⁻¹⁹ C

t = 5.0 × 10⁻⁴ m

Starting with:

V_H = IB/(n|q|t)

Rearrange:

B = V_Hn|q|t/I

Substitute:

B = [(2.0 × 10⁻³)(2.5 × 10²²)(1.60 × 10⁻¹⁹)(5.0 × 10⁻⁴)]/(0.040)

This gives:

B = 0.10 T

---

Example: Finding Carrier Density

Suppose:

V_H = 4.0 × 10⁻³ V

I = 0.020 A

B = 0.40 T

t = 5.0 × 10⁻⁴ m

|q| = 1.60 × 10⁻¹⁹ C

Use:

V_H = IB/(n|q|t)

Rearrange:

n = IB/(V_H|q|t)

Substituting the values gives:

n = 2.5 × 10²² m⁻³

This shows how Hall measurements can be used to investigate the microscopic properties of materials.

---

Designing a Sensitive Hall Sensor

From:

V_H = IB/(n|q|t)

a larger Hall voltage can be produced by:

• increasing current I
• increasing magnetic field B
• using a material with lower carrier density n
• decreasing Hall-element thickness t

However, engineering constraints matter.

Increasing current too much may cause:

heating

Making the sensor extremely thin may reduce:

mechanical strength

Choosing a material therefore involves balancing sensitivity with reliability.

---

Limitations and Measurement Errors

Hall measurements can be affected by:

• incorrect probe orientation
• temperature changes
• electrical noise
• imperfect calibration
• nearby magnetic fields
• sensor offset
• limited sensor range

A good measurement procedure should therefore include:

• calibration
• correct alignment
• repeated measurements
• appropriate range selection
• control of nearby magnetic sources

---

Common Mistakes

Mistake: Saying the Hall Effect occurs because charges are attracted to a magnet.
The sideways separation is caused by magnetic force on moving charge carriers.

Mistake: Saying a magnetic field creates the current.
In the basic Hall setup, a current is already flowing. The magnetic field deflects its charge carriers.

Mistake: Saying the Hall voltage develops along the same direction as the current.
The Hall voltage develops across the conductor, transverse to the current.

Mistake: Forgetting that magnetic force depends on particle motion.
Stationary charges do not experience magnetic force.

Mistake: Assuming charge continues accumulating indefinitely.
The growing Hall electric field eventually balances the magnetic force.

Mistake: Saying stronger B produces smaller Hall voltage.
For fixed conditions, V_H ∝ B.

Mistake: Saying thicker Hall elements produce larger Hall voltage.
For the standard geometry, V_H ∝ 1/t.

Mistake: Ignoring probe orientation.
A Hall probe responds to the appropriate component of the magnetic field.

Mistake: Assuming Hall sensors can only measure magnetic fields.
They can also be used for position, speed, current and proximity sensing.

---

Key Terms

Hall Effect — production of a transverse voltage when moving charge carriers in a conductor or semiconductor are deflected by a magnetic field.

Hall voltage (V_H) — potential difference produced across a Hall element by charge separation.

Hall field (E_H) — electric field created across the material by separated charges.

Charge carrier — mobile charged particle responsible for electric current, such as an electron or hole.

Drift velocity (v_d) — average directed velocity of charge carriers due to an applied electric field.

Carrier density — number of mobile charge carriers per unit volume.

Hall coefficient (R_H) — material quantity related to carrier density and charge, with R_H = 1/(nq) for a simple one-carrier model.

Hall probe — instrument using the Hall Effect to measure magnetic flux density.

Hall sensor — electronic device using Hall voltage to detect magnetic fields.

Analogue Hall sensor — Hall sensor producing an output related continuously to magnetic-field strength.

Digital Hall sensor — Hall sensor producing switching or digital output when specified magnetic conditions are reached.

Contactless sensing — detecting position or motion without direct mechanical contact.

---

Key Takeaways

• The Hall Effect occurs when moving charge carriers are deflected by a magnetic field.
• The magnetic force causes charge to accumulate on opposite sides of a conductor or semiconductor.
• This charge separation produces a Hall voltage.
• The growing Hall electric force eventually balances the magnetic force.
• For perpendicular motion, F_B = |q|vB.
• At equilibrium, E_H = v_dB.
• A useful Hall-voltage relationship is V_H = IB/(nqt), with sign depending on carrier type and geometry.
• Hall voltage increases with current.
• Hall voltage increases with magnetic flux density.
• Hall voltage increases as carrier density decreases.
• Hall voltage increases as Hall-element thickness decreases.
• Semiconductors are particularly useful for Hall sensors because their carrier densities can be relatively low.
• Hall measurements can provide information about charge-carrier type and density.
• A Hall probe converts Hall voltage into a measurement of magnetic flux density.
• Correct probe orientation is important.
• Reversing the magnetic field reverses Hall-voltage polarity.
• Hall sensors can detect position and proximity.
• Hall sensors can measure rotational speed.
• Hall sensors can help control brushless motors.
• Hall sensors can measure electric current indirectly by detecting the magnetic field produced by the current.
• Hall Effect technology is widely used in vehicles, electronics, robotics, industrial control and scientific measurement.

Check Your Understanding

1. What is the Hall Effect?
2. Why do moving charge carriers become separated in a Hall element?
3. What is Hall voltage?
4. In what direction is Hall voltage produced relative to the main current?
5. Explain why charge does not continue accumulating indefinitely on one side of the Hall element.
6. What two forces balance when a steady Hall voltage has developed?
7. Show how E_H = v_dB is obtained for perpendicular motion.
8. State the relationship V_H = IB/(nqt) and identify each quantity.
9. How does increasing current affect Hall voltage?
10. How does increasing magnetic flux density affect Hall voltage?
11. How does increasing carrier density affect Hall voltage?
12. How does decreasing the Hall element's thickness affect Hall voltage?
13. Why are semiconductors particularly useful for Hall Effect sensors?
14. Explain how a Hall probe can measure magnetic flux density.
15. Why is the orientation of a Hall probe important?
16. A Hall sensor produces 15 mV in a 0.20 T field. Assuming a linear response, what voltage would it produce in a 0.60 T field?
17. A wheel containing one magnet produces 30 Hall-sensor pulses per second. Calculate its rotational speed in rpm.
18. Explain how Hall Effect sensors can be used in a brushless DC motor.
19. Explain how a Hall sensor can measure electric current without being directly connected in series with the main circuit.
20. Describe three practical applications of Hall Effect sensors and explain what the sensor detects in each case.

Mass Spectrometers and Particle Physics

What Is a Mass Spectrometer?

A mass spectrometer is an instrument used to identify and separate charged particles according to their mass-to-charge ratio.

The mass-to-charge ratio is written:

m/q

or, in mass spectrometry, commonly:

m/z

where z represents the ion's charge number.

Mass spectrometers allow scientists to determine:

• which isotopes are present in a sample
• relative isotope abundances
• atomic and molecular masses
• chemical composition
• properties of unknown substances
• masses and identities of particles produced in physics experiments

The central idea is that electric and magnetic fields can control the motion of charged particles.

---

Why the Particles Must Be Charged

Electric and magnetic fields exert forces on charged particles.

For an electric field:

F_E = qE

For a moving particle in a magnetic field:

F_B = |q|vB sin θ

A neutral atom or molecule has:

q = 0

so it cannot be controlled directly by these forces in the same way.

Therefore, samples are typically converted into:

ions

before their mass-to-charge ratios are analyzed.

---

The Basic Stages of Mass Spectrometry

Different mass spectrometers use different designs, but a simplified system may contain:

1. Ionization
2. Acceleration
3. Velocity selection or other ion control
4. Deflection/separation
5. Detection
6. Data analysis

The details vary greatly between modern instruments, but these stages provide a useful model for understanding the physics.

---

Ionization

The first step is to produce charged particles.

Atoms or molecules may lose or gain electrons.

For example:

X → X⁺ + e⁻

The resulting positive ion can then respond to electric and magnetic fields.

An ion with one missing electron has charge:

q = +e

where:

e = 1.60 × 10⁻¹⁹ C

An ion with two missing electrons has:

q = +2e

The ion's charge is important because mass spectrometry separates particles according to:

m/q

not mass alone.

---

Acceleration by an Electric Field

Electric fields can accelerate ions.

If an ion of charge magnitude |q| is accelerated from rest through a suitable potential difference V:

|q|V = ½mv²

Therefore:

v = √(2|q|V/m)

This converts electrical potential energy into kinetic energy.

Electric fields are therefore useful for controlling the energy and speed of charged particles.

---

Example: Accelerating an Ion

A singly charged positive ion is accelerated from rest through:

V = 2000 V

Suppose:

m = 3.32 × 10⁻²⁶ kg

q = 1.60 × 10⁻¹⁹ C

Use:

|q|V = ½mv²

v = √(2|q|V/m)

v = √[(2)(1.60 × 10⁻¹⁹)(2000)/(3.32 × 10⁻²⁶)]

v ≈ 1.39 × 10⁵ m/s

---

The Velocity Selector

Some classic mass-spectrometer designs use a velocity selector.

A velocity selector contains crossed:

• electric field E
• magnetic field B

The fields are arranged so that the electric and magnetic forces act in opposite directions.

Electric force:

F_E = |q|E

Magnetic force:

F_B = |q|vB

for perpendicular motion.

For an undeflected particle:

F_E = F_B

Therefore:

|q|E = |q|vB

Cancel |q|:

v = E/B

Only particles travelling at this speed pass straight through.

---

Why a Velocity Selector Is Useful

Suppose different ions enter the magnetic analyzing region with many different speeds.

Their paths would depend on both:

• mass
• charge
• speed

This would make interpretation more difficult.

A velocity selector allows only ions with approximately the same selected speed to continue.

Then differences in trajectory depend mainly on:

mass-to-charge ratio

for a fixed magnetic field.

---

Magnetic Deflection

After velocity selection, ions may enter another magnetic field.

If the ions enter perpendicular to the field:

F_B = |q|vB

The magnetic force is always perpendicular to the velocity.

Therefore, it acts as a centripetal force.

The ions follow circular paths.

---

Deriving the Circular Radius

Magnetic force:

F_B = |q|vB

Centripetal force:

F_c = mv²/r

Set them equal:

|q|vB = mv²/r

Cancel one v:

|q|B = mv/r

Rearrange:

r = mv/(|q|B)

This is one of the most important equations for charged-particle motion in a magnetic field.

---

What Determines the Radius?

From:

r = mv/(|q|B)

we can see:

r ∝ m

Greater mass → larger radius

r ∝ v

Greater speed → larger radius

r ∝ 1/|q|

Greater charge magnitude → smaller radius

r ∝ 1/B

Stronger magnetic field → smaller radius

---

Mass-to-Charge Ratio

Rearrange:

r = mv/(|q|B)

to obtain:

m/|q| = Br/v

Therefore, if B, r and v are known, the particle's mass-to-charge ratio can be determined.

This is the fundamental measurement made by many forms of mass spectrometry.

---

Combining with a Velocity Selector

The velocity selector gives:

v = E/B₁

Suppose the analyzing magnetic field is B₂.

Then:

r = mv/(|q|B₂)

Substitute:

v = E/B₁

to obtain:

r = mE/(|q|B₁B₂)

If the same magnetic-field strength B is used in both regions:

r = mE/(|q|B²)

Therefore:

r ∝ m/|q|

when E and B are fixed.

---

Comparing Two Ions

Suppose two ions:

• have the same charge
• have the same speed
• enter the same magnetic field

Then:

r = mv/(|q|B)

so:

r ∝ m

The heavier ion follows a path with a:

larger radius

The lighter ion follows a path with a:

smaller radius

---

Example: Two Isotopes

Suppose two singly charged ions are isotopes of the same element.

Ion A:

mass = 20 u

Ion B:

mass = 22 u

Both have the same speed and enter the same magnetic field.

Then:

r_B/r_A = m_B/m_A

r_B/r_A = 22/20

r_B/r_A = 1.10

Therefore, the heavier isotope follows a circular path with a radius about:

10% larger

---

What Are Isotopes?

Isotopes are atoms of the same element with:

• the same number of protons
• different numbers of neutrons

Therefore, isotopes have:

• the same atomic number
• different mass numbers

For example, carbon includes:

• carbon-12
• carbon-13
• carbon-14

Their different masses allow them to be distinguished using mass spectrometry.

---

Separating Isotopes

Suppose two isotopes are ionized so that each has charge:

+e

They pass through the same velocity selector.

Therefore, they have the same speed.

They then enter the same magnetic field.

Since:

r = mv/(|q|B)

and v, |q| and B are identical:

r ∝ m

Therefore:

heavier isotope → larger radius

lighter isotope → smaller radius

The ions reach different locations on the detector.

---

Example: Neon Isotopes

Naturally occurring neon contains several isotopes, particularly:

• neon-20
• neon-21
• neon-22

If these atoms are ionized to the same charge and analyzed under the same conditions, their different masses produce different:

m/q values

They can therefore be detected separately.

The instrument can determine both:

• isotope identity
• relative abundance

---

Mass Spectrum

The output of a mass spectrometer is often represented as a:

mass spectrum

A mass spectrum plots signal intensity against:

mass-to-charge ratio, m/z

Each peak corresponds to ions with a particular m/z value.

The height or intensity of a peak indicates how much of that ion was detected relative to others.

For isotope measurements, this allows both isotope masses and relative abundances to be determined.

---

Reading an Isotope Mass Spectrum

Suppose an element produces two major peaks:

m/z = 35

and:

m/z = 37

If the ions are singly charged, these may correspond to isotopes with mass numbers approximately:

35 and 37.

If the peak at 35 is much larger, the isotope with mass 35 is more abundant.

Thus:

peak position → m/z

peak intensity → relative abundance

---

Relative Atomic Mass

Mass spectrometry can be used to calculate the average relative atomic mass of an element from its isotope abundances.

For example, suppose an element contains:

75% isotope-35

25% isotope-37

Then:

relative atomic mass = (35 × 0.75) + (37 × 0.25)

= 26.25 + 9.25

= 35.5

This explains why many relative atomic masses on the periodic table are not whole numbers.

---

Example: Isotope Abundance

Suppose an element has:

80% isotope-10

20% isotope-11

Average mass:

= (10 × 0.80) + (11 × 0.20)

= 8.0 + 2.2

= 10.2

Mass spectrometry provides the isotope data needed for calculations like this.

---

Charge State Matters

Mass spectrometers measure:

mass-to-charge ratio

not simply mass.

Consider an ion with mass:

40 u

and charge:

+1

Its approximate m/z is:

40

But if the same mass carried charge:

+2

its approximate m/z would be:

20

Therefore, charge state must be considered when interpreting mass spectra.

---

Why m/z Is Used

In practical mass spectrometry, the horizontal axis is usually labelled:

m/z

where:

• m represents ion mass
• z represents charge number

For a singly charged ion:

z = 1

so the m/z value corresponds closely to the ion mass in the conventional units used in mass spectrometry.

For multiply charged ions, the same particle can appear at a smaller m/z value.

---

Magnetic Field Does Not Change Particle Speed

The magnetic force acts perpendicular to the particle's velocity.

Therefore, a magnetic field acting alone:

• changes velocity direction
• does not change speed
• does not change kinetic energy

This makes magnetic fields especially useful for:

steering and separating particle beams

Electric fields, by contrast, can do work on charged particles and change their kinetic energy.

---

Electric Fields vs Magnetic Fields in Mass Spectrometry

Together, these fields provide powerful control over charged particles.

---

Example: Finding the Radius

A singly charged positive ion has:

m = 3.32 × 10⁻²⁶ kg

v = 2.0 × 10⁵ m/s

B = 0.50 T

q = 1.60 × 10⁻¹⁹ C

Use:

r = mv/(|q|B)

r = [(3.32 × 10⁻²⁶)(2.0 × 10⁵)] / [(1.60 × 10⁻¹⁹)(0.50)]

r = 0.083 m

Therefore:

r ≈ 8.3 cm

---

Example: Finding Mass

An ion travels at:

v = 3.0 × 10⁵ m/s

through:

B = 0.40 T

Its circular path has:

r = 0.20 m

The ion is singly charged:

|q| = 1.60 × 10⁻¹⁹ C

From:

r = mv/(|q|B)

rearrange:

m = |q|Br/v

m = [(1.60 × 10⁻¹⁹)(0.40)(0.20)]/(3.0 × 10⁵)

m ≈ 4.27 × 10⁻²⁶ kg

---

Example: Finding Mass-to-Charge Ratio

A particle follows a circular path with:

r = 0.15 m

in:

B = 0.80 T

at:

v = 4.0 × 10⁵ m/s

Use:

m/|q| = Br/v

m/|q| = (0.80)(0.15)/(4.0 × 10⁵)

m/|q| = 3.0 × 10⁻⁷ kg/C

---

Multi-Step Mass Spectrometer Problem

A velocity selector contains:

E = 6.0 × 10⁴ N/C

B₁ = 0.20 T

An ion passes through undeflected.

It then enters a second magnetic field:

B₂ = 0.50 T

and follows a circular path:

r = 0.10 m

The ion has charge:

|q| = 1.60 × 10⁻¹⁹ C

Find the speed

v = E/B₁

v = (6.0 × 10⁴)/(0.20)

v = 3.0 × 10⁵ m/s

Find the mass

Use:

m = |q|B₂r/v

m = [(1.60 × 10⁻¹⁹)(0.50)(0.10)]/(3.0 × 10⁵)

m ≈ 2.67 × 10⁻²⁶ kg

This demonstrates how electric and magnetic fields can be combined to determine an unknown particle mass.

---

From Mass Spectrometers to Particle Physics

The same principles used in a classroom mass spectrometer are important in particle physics.

Particle physicists need to determine properties such as:

• charge
• momentum
• mass
• energy
• particle identity

Charged particles moving through magnetic fields follow curved trajectories.

By studying these trajectories, scientists can reconstruct important properties of the particles.

---

Particle Momentum

For a charged particle moving perpendicular to a magnetic field:

r = mv/(|q|B)

For non-relativistic motion:

p = mv

Therefore:

p = |q|Br

More generally, the relationship between momentum and curvature remains fundamental in relativistic particle detectors, with the appropriate relativistic momentum.

Therefore:

large radius → larger momentum for fixed |q| and B

small radius → smaller momentum

---

Charge Sign from Curvature

Positive and negative particles curve in opposite directions in the same magnetic field.

Therefore, the direction of curvature can reveal the:

sign of the particle's charge

For example, two particles leaving a collision point and curving in opposite directions may carry opposite charges.

This is extremely useful in particle identification.

---

Particle Detectors

Modern particle detectors can contain:

• tracking detectors
• powerful magnets
• calorimeters
• particle-identification systems
• muon detectors

Tracking detectors record the paths of charged particles.

The magnetic field bends these paths.

From the curvature, scientists can determine information about:

• momentum
• charge sign
• particle interactions

---

High-Momentum Particles

From:

p = |q|Br

for a fixed magnetic field and charge magnitude:

larger momentum → larger radius

A very high-momentum particle bends only slightly.

A lower-momentum particle bends more strongly.

Therefore:

straighter-looking track → generally greater momentum

under otherwise comparable conditions.

---

Particle Collisions

In high-energy particle physics, beams of particles can be accelerated and collided.

The collisions may produce many new particles.

These particles move outward through detectors.

Charged particles curve in magnetic fields.

Neutral particles do not leave curved charged-particle tracks directly, although their presence may be inferred from:

• decay products
• energy deposits
• missing momentum or energy
• other detector signatures

---

Mass Spectrometry and Particle Detection: The Connection

Traditional mass spectrometry and particle-physics detectors share several fundamental ideas:

Both rely on the predictable motion of charged particles in electromagnetic fields.

---

Particle Accelerators

Particle accelerators use electric fields to increase particle energy.

Magnetic fields are then used to:

• steer beams
• bend trajectories
• focus particles
• maintain beam paths

This reflects the same basic distinction seen in mass spectrometry:

electric fields → change particle energy

magnetic fields → control particle direction

---

Mass Spectrometry in Modern Science

Mass spectrometry has applications far beyond basic isotope identification.

It is used in:

• chemistry
• environmental science
• geology
• astronomy
• medicine
• pharmaceuticals
• forensic science
• materials science
• nuclear physics
• particle physics

Scientists can use mass spectrometric techniques to identify extremely small quantities of substances and make very precise measurements of particle masses.

---

Dating and Isotope Studies

Mass spectrometers can precisely measure isotope ratios.

These measurements are useful in:

• radiometric dating
• geological studies
• climate reconstruction
• tracing chemical sources
• nuclear research

For example, isotope ratios can reveal information about the origin and history of materials.

---

Detecting Unknown Substances

Molecules can be ionized and sometimes broken into fragments.

Different fragments have different m/z values.

The resulting pattern can act like a:

molecular fingerprint

Scientists compare the pattern with known data to help identify an unknown substance.

This is widely used in analytical chemistry and forensic science.

---

Mass Spectrometers vs Particle Accelerators

A mass spectrometer is mainly designed to:

identify or analyze particles according to m/z

A particle accelerator is mainly designed to:

increase particle energy and control particle beams

However, both use many of the same electromagnetic principles.

The technologies therefore overlap strongly in their underlying physics.

---

Analyzing a Mass Spectrometer Problem

A useful strategy is:

Identify the stage

Is the particle being:

• accelerated?
• velocity-selected?
• magnetically deflected?

Choose the relevant equation

Acceleration:

|q|V = ½mv²

Velocity selector:

v = E/B

Magnetic deflection:

r = mv/(|q|B)

Check the charge

Remember that the device measures:

m/q

not simply m.

Check units

Use:

• kg for mass
• C for charge
• m/s for speed
• T for magnetic flux density
• m for radius
• V for potential difference

---

Equation Summary

Electric force

F_E = qE

Magnetic force magnitude

F_B = |q|vB sin θ

Magnetic force for perpendicular motion

F_B = |q|vB

Energy gained through a suitable potential difference

|q|V = ½mv²

Velocity selector

v = E/B

Circular radius

r = mv/(|q|B)

Mass-to-charge ratio

m/|q| = Br/v

Momentum from curvature

p = |q|Br

These equations connect the major ideas of charged-particle motion, mass spectrometry and particle detection.

---

Common Mistakes

Mistake: Saying a mass spectrometer separates particles only according to mass.
It responds to mass-to-charge ratio.

Mistake: Forgetting that particles must be charged.
Electric and magnetic fields require charged particles for direct electromagnetic control.

Mistake: Saying magnetic fields increase particle speed.
A magnetic field acting alone changes direction, not speed.

Mistake: Assuming heavier ions curve more strongly.
For equal q, v and B, heavier ions have a larger radius and therefore curve less sharply.

Mistake: Ignoring ion charge.
A doubly charged ion behaves differently from a singly charged ion of the same mass.

Mistake: Assuming m/z always equals mass.
This is approximately true for many singly charged ions when conventional mass-spectrometry units are used, but not for multiply charged ions.

Mistake: Confusing peak position with abundance.
Peak position indicates m/z; peak intensity indicates relative signal or abundance under the relevant measurement conditions.

Mistake: Saying all modern mass spectrometers use exactly the same magnetic-sector design.
Many modern instruments use different methods of separating and detecting ions. The magnetic-sector model is especially useful for understanding the underlying charged-particle physics.

---

Key Terms

Mass spectrometer — instrument used to separate and identify ions according to mass-to-charge ratio.

Mass-to-charge ratio (m/q) — ratio of particle mass to electric charge.

m/z — conventional mass-spectrometry notation for mass-to-charge ratio using the ion's charge number.

Ionization — process of producing charged particles from neutral atoms or molecules.

Ion — electrically charged atom or molecule.

Velocity selector — crossed electric and magnetic fields that allow particles of a selected speed to pass undeflected.

Magnetic deflection — bending of a charged-particle trajectory by a magnetic field.

Isotope — atom of an element with the same proton number but a different neutron number.

Mass spectrum — representation of detected ion signal against m/z.

Relative abundance — amount of one isotope or ion relative to others detected in the sample.

Detector — component that records arriving particles or ions.

Particle accelerator — device using electromagnetic fields to accelerate and control charged particles.

Tracking detector — detector used to reconstruct charged-particle trajectories.

Momentum — quantity related to mass and velocity; for charged-particle curvature in a magnetic field, p = |q|Br for perpendicular motion.

---

Key Takeaways

• A mass spectrometer analyzes charged particles according to mass-to-charge ratio.
• Neutral atoms or molecules must generally be ionized before electromagnetic analysis.
• Electric fields can accelerate ions and change their kinetic energy.
• Magnetic fields bend charged-particle trajectories.
• A velocity selector can produce particles with a selected speed using v = E/B.
• A particle moving perpendicular to a magnetic field can follow a circular path.
• The radius is r = mv/(|q|B).
• Greater mass produces a larger radius when q, v and B are unchanged.
• Greater charge magnitude produces a smaller radius.
• Stronger magnetic fields produce smaller radii.
• Mass spectrometry can separate isotopes because isotopes have different masses.
• Peak position in a mass spectrum identifies m/z.
• Peak intensity can provide information about relative abundance.
• Isotope abundances can be used to calculate average relative atomic mass.
• Mass spectrometers measure m/z rather than mass alone.
• Multiply charged ions can have much smaller m/z values than singly charged ions of the same mass.
• Magnetic curvature can be used to determine charged-particle momentum.
• Positive and negative particles curve in opposite directions in the same magnetic field.
• Particle accelerators and mass spectrometers use many of the same electromagnetic principles.
• Modern particle physics relies heavily on electromagnetic control and detection of charged particles.

Check Your Understanding

1. What is the main purpose of a mass spectrometer?
2. Why must particles normally be ionized before they can be analyzed using electric and magnetic fields?
3. What is meant by mass-to-charge ratio?
4. Describe the main stages of a simplified mass spectrometer.
5. Explain how an electric field can accelerate an ion.
6. State the energy equation for an ion accelerated from rest through a suitable potential difference.
7. Explain how a velocity selector works.
8. Derive v = E/B for an undeflected particle.
9. Why is selecting particles with the same speed useful before magnetic deflection?
10. Explain why a charged particle can follow a circular path in a magnetic field.
11. State the equation r = mv/(|q|B) and explain each quantity.
12. Two singly charged ions have the same speed in the same magnetic field. Which follows the larger-radius path: the heavier or lighter ion?
13. Explain how isotopes can be separated using a mass spectrometer.
14. What information does the position of a peak in a mass spectrum provide?
15. What information can peak intensity provide?
16. An element contains 60% isotope-10 and 40% isotope-11. Calculate its average relative atomic mass.
17. A particle has m = 4.0 × 10⁻²⁶ kg, q = 1.60 × 10⁻¹⁹ C, v = 2.0 × 10⁵ m/s and enters a 0.50 T magnetic field perpendicularly. Calculate its circular radius.
18. Explain why a +2 ion behaves differently from a +1 ion of the same mass and speed in the same magnetic field.
19. Explain how the curvature of a charged-particle track can provide information about its momentum.
20. Describe two important connections between mass spectrometry and modern particle-physics research.