Hall effect

CIE A-Level Physics· Unit 23: Magnetic fields· 25 min read

1. Origin of the Hall Effect★★☆☆☆⏱ 8 min

When a current-carrying conductor is placed in a magnetic field perpendicular to the direction of current flow, moving charge carriers experience a Lorentz force that deflects them toward one side of the conductor.

📘 Definition

Hall Effect

The generation of a transverse potential difference (Hall voltage) across a current-carrying conductor placed in a perpendicular magnetic field, due to charge separation caused by the Lorentz force on moving charge carriers.

Example:

A thin copper strip carrying current placed between the poles of a permanent magnet develops a small voltage across its width.

Equilibrium is reached when the magnetic Lorentz force on the charge carriers is balanced by the electric force from the separated charge: $F_{B} = F_{E}$

Bqv = \frac{qV_H}{w}

Where $B$ = magnetic flux density, $q$ = charge of one carrier, $v$ = drift velocity of carriers, $V_{H}$ = Hall voltage, $w$ = width of the conductor across which the voltage develops.

📐 Worked Example

A copper strip of width 2.0 cm has electrons moving with drift velocity $3.2 \times 1 0^{- 4} m s^{- 1}$ in a 0.5 T magnetic field. What is the electric field strength across the strip at equilibrium?

1
At equilibrium, magnetic force equals electric force, so cancel the charge term $q$ from both sides:
2
$E = Bv$
3
Substitute the given values to get the electric field strength:
4
$E = (0.5)(3.2 \times 10^{-4}) = 1.6 \times 10^{-4} \text{ V m}^{-1}$

2. Derivation of the Hall Voltage Equation★★★☆☆⏱ 10 min

🔬 Derivation

Goal:

Derive an expression for Hall voltage $V_{H}$ in terms of measurable quantities $B$ , $I$ , $n$ , $q$ , and $t$ .

Starting from:

Equilibrium condition $B q v = q V_{H} / w$ and current $I = n A v q$

1
Rearrange the equilibrium condition to get: $V_{H} = B v w$
2
Cross-sectional area $A = w t$ , where $t$ is the thickness of the conductor parallel to the magnetic field. Substitute into the current equation:
3
$I = n (w t) v q$ , rearrange to solve for $v$ :
4
$v = \frac{I}{nwtq}$
5
Substitute $v$ into the expression for $V_{H}$ :
6
$V_H = B \left(\frac{I}{nwtq}\right) w$

Result:

The width term $w$ cancels out, giving the final Hall voltage formula:

V_H = \frac{BI}{nqt}

📐 Worked Example

A Hall probe made from n-type semiconductor has charge carrier density $n = 1.0 \times 1 0^{20} m^{- 3}$ and thickness $t = 0.50 mm$ . A current of 10 mA flows through the probe. Calculate the Hall voltage in a 0.2 T magnetic field ( $e = 1.6 \times 1 0^{- 19} C$ ).

1
Convert all quantities to SI units:
2
$t = 0.50 \text{ mm} = 0.50 \times 10^{-3} \text{ m}, \quad I = 10 \text{ mA} = 0.010 \text{ A}$
3
Substitute into the Hall voltage formula:
4
$V_H = \frac{BI}{net}$
5
Plug in the values:
6
$V_H = \frac{(0.2)(0.01)}{(1.0 \times 10^{20})(1.6 \times 10^{-19})(0.5 \times 10^{-3})} = \frac{0.002}{8 \times 10^{-3}} = 0.25 \text{ V}$

3. Applications and Key Properties★★★☆☆⏱ 7 min

The Hall effect has many practical applications, thanks to the linear relationship between Hall voltage and magnetic flux density. Two of the most important are:

Hall probes for magnetic measurement: If $n$ , $q$ , $t$ , and $I$ are fixed for the probe, $V_{H} \propto B$ , so measuring $V_{H}$ directly gives the magnetic flux density.
Identifying semiconductor type: The sign of the Hall voltage corresponds to the sign of the majority charge carriers, distinguishing n-type (electrons) from p-type (holes) semiconductors.

📐 Worked Example

Current flows left to right along a semiconductor strip, with magnetic field directed into the plane of the strip. The top edge of the strip becomes positively charged. Is the semiconductor n-type or p-type?

1
For p-type semiconductors, majority charge carriers are positive holes that move in the same direction as conventional current (left to right).
2
Use Fleming's Left Hand Rule for force on positive charge: First finger (field) into page, second finger (current) left to right, thumb points upwards.
3
Positive holes are deflected upwards, so the top edge accumulates positive charge, matching the observation.

4. Common Pitfalls

Wrong move:

Confusing thickness $t$ and width $w$ in the Hall voltage formula

Why:

The formula uses thickness along the magnetic field direction, not the width across which Hall voltage is measured

Correct move:

Remember $V_{H} = B I / (n q t)$ , where $t$ is the dimension parallel to the magnetic field, not transverse.

Wrong move:

Treating holes as electrons moving opposite when finding deflection direction

Why:

This leads to the wrong sign of Hall voltage because holes are positive charge carriers, not negative electrons

Correct move:

Apply Fleming's Left Hand Rule directly to the charge of the majority carrier, not electron flow opposite to current.

Wrong move:

Forgetting to convert prefixed units to SI units before calculation

Why:

The Hall voltage formula is derived for SI units, so leaving mm or mA un-converted gives wrong orders of magnitude

Correct move:

Always convert all quantities to metres, amperes, and tesla before calculating $V_{H}$ .

Wrong move:

Assuming Hall effect only occurs in metals

Why:

The Hall effect occurs in any material with moving charge carriers, but is much weaker in metals than semiconductors

Correct move:

Recognize that low carrier density in semiconductors produces large measurable Hall voltages, so semiconductors are used for Hall probes.

5. Quick Reference Cheatsheet

Quantity/Rule	Symbol	Relation
Equilibrium condition	$F_{B} = F_{E}$	$B q v = \frac{q V _{H}}{w}$
Hall voltage formula	$V_{H}$	$V_{H} = \frac{B I}{n q t}$
Proportionality for probes	$V_{H} \propto B$	Linear for constant current $I$
Positive Hall voltage sign		Indicates p-type semiconductor
N	e	g	a	t	i	v	e		H	a	l	l		v	o	l	t	a	g	e		s	i	g	n

I	n	d	i	c	a	t	e	s		n		t	y	p	e		s	e	m	i	c	o	n	d	u	c	t	o	r

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 22
Calculate Hall voltage for copper foil
2021 · 13
Explain Hall effect in p-type semiconductors
2023 · 21
Derive Hall voltage formula

Going deeper

simulationPhET Hall Effect SimulationInteractive visualization of charge separation