Experimental Physics & Hall Effect: Theory, Devices & Applications

TAKEAWAYS

• The Hall effect generates a voltage perpendicular to current and magnetic field, enabling non-contact current/field measurements (used in eSewa’s payment terminals and Ncell’s Hall-effect sensors).
• Magnetic dipole moments explain atomic/molecular behavior (e.g., why iron aligns in magnets) and are critical in MRI machines and hard drives.
• Fermi energy ($E_F$) and effective mass ($m^{\ast }$) determine semiconductor properties (e.g., why aluminum’s $E_F=12\text{eV}$ makes it conductive).
• Torque on current loops ($\tau =NIAB\sin \theta$) is used in electric meters and motor design (e.g., Pathao’s delivery drones use torque for stability).
• Experimental techniques (e.g., Hall probes, Fermi energy calculations) are foundational for NEPSE’s stock market sensors and bank card readers.

1. The Hall Effect: Theory and Derivation

What is the Hall Effect?

When a current-carrying conductor (or semiconductor) is placed in a magnetic field perpendicular to its length, a voltage ($V_H$) develops across its width. This is called the Hall voltage, caused by the Lorentz force deflecting moving charges.

graph LR
    A["Current (I) flows left"] --> B["Magnetic Field (B) into page"]
    B --> C["Lorentz Force (F = qvB) pushes electrons up"]
    C --> D["Electron buildup creates Hall voltage (V_H) across width"]
    D --> E["Steady state: Electric force balances Lorentz force"]

Key Equations

1. Hall Voltage: $$V_H=\frac{IB}{nqt}$$

   • $I$: Current (A)
   • $B$: Magnetic field (T)
   • $n$: Charge carrier density (m⁻³)
   • $q$: Charge of carrier ($1.6\times 10^{-19}\text{C}$ for electrons)
   • $t$: Thickness of slab (m)

2. Hall Coefficient ($R_H$): $$R_H=\frac{V_{H}t}{IB}=\frac{1}{nq}$$

   • Positive $R_H$: Holes (p-type semiconductors)
   • Negative $H_R$: Electrons (n-type semiconductors)

Real-World Example: eSewa’s Payment Terminals

eSewa uses Hall-effect sensors to:

• Measure current in payment terminals (ensuring correct transaction processing).
• Detect magnetic card strips (older eSewa cards used Hall sensors for authentication).
• Why? Hall sensors are non-contact, durable, and immune to EMI (unlike resistive sensors).

A cross-section of a Hall probe showing current, magnetic field, and Hall voltage generation. (Image: Dracheschreck, CC BY-SA 3.0, via Wikimedia Commons)

Worked Example: Copper Slab in a Magnetic Field

A copper slab (thickness $t=0.5\text{cm}=0.005\text{m}$, width $w=2\text{cm}=0.02\text{m}$) carries $I=50\text{A}$ in a $B=1.5\text{T}$ field perpendicular to its plane. Copper has $n=8.5\times 10^{28}{\text{m}}^{-3}$. Calculate:

1. Hall voltage ($V_H$).
2. Hall coefficient ($R_H$).
3. Direction of $V_H$ (for electrons vs. holes).

Solution:

1. Hall Voltage: $$V_H=\frac{IB}{nqt}=\frac{(50)(1.5)}{(8.5\times 10^{28})(1.6\times 10^{-19})(0.005)}=1.10\times 10^{-4}\text{V}=0.11\text{mV}$$ This tiny voltage is measurable with sensitive amplifiers.

2. Hall Coefficient: $$R_H=\frac{1}{nq}=\frac{1}{(8.5\times 10^{28})(1.6\times 10^{-19})}=-7.35\times 10^{-5}{\text{m}}^3/\text{C}$$ Negative sign confirms electrons are majority carriers.

3. Direction:

   • For electrons (copper): $V_H$ is upward (opposite to Lorentz force).
   • For holes (e.g., silicon): $V_H$ would be downward.

2. Applications of the Hall Effect

3. Magnetic Dipole Moment: Atomic and Molecular Effects

Definition

A magnetic dipole moment ($\mu$) arises from:

• Orbital motion of electrons (current loop).
• Spin of electrons/protons.

For a current loop: $$\mu =IA$$

• $I$: Current (A)
• $A$: Area of loop (m²)

For an atom: $$\mu ={\mu }_B\sqrt{j(j+1)}$$

• ${\mu }_B=9.27\times 10^{-24}\text{J/T}$ (Bohr magneton)
• $j$: Total angular momentum quantum number

Effects on Atoms and Molecules

1. Zeeman Effect:

   • Splitting of spectral lines in a magnetic field (used in MRI machines).
   • Example: Hydrogen’s $n=2$ to $n=1$ transition splits into 3 lines.

2. Diamagnetism vs. Paramagnetism:

   • Diamagnetic (e.g., copper): $\mu$ opposes $B$ (weak, temporary).
   • Paramagnetic (e.g., aluminum): $\mu$ aligns with $B$ (stronger, permanent).

3. Ferromagnetism (e.g., iron):

   • Domains align spontaneously, creating permanent magnets (used in hard drives).

Worked Example: Torque on a Current Loop (Ncell Antenna)

An Ncell antenna loop has:

• Area $A=0.01{\text{m}}^2$
• Current $I=2\text{A}$
• Magnetic field $B=5\times 10^{-5}\text{T}$ (Earth’s field)
• Angle $\theta =30^{\circ }$ to $B$

Calculate:

1. Magnetic dipole moment ($\mu$).
2. Torque ($\tau$) acting on the loop.

Solution:

1. Dipole Moment: $$\mu =IA=(2)(0.01)=0.02\text{A}\cdot {\text{m}}^2$$

2. Torque: $$\tau =\mu B\sin \theta =(0.02)(5\times 10^{-5})(\sin 30^{\circ })=5\times 10^{-7}\text{N}\cdot \text{m}$$ This torque tries to align the loop with Earth’s field (why antennas auto-rotate in storms!).

4. Fermi Energy and Effective Mass in Solids

Fermi Energy ($E_F$)

The highest occupied energy level at 0 K in a metal/semiconductor. For a free-electron gas: $$E_F=\frac{{\hslash }^2}{2m}{\left(3{\pi }^2\frac{N}{V}\right)}^{2/3}$$

• $N/V$: Electron density (m⁻³)
• $m$: Electron mass ($9.11\times 10^{-31}\text{kg}$)

Worked Example: Aluminum’s Fermi Energy

Given:

• Density ($\rho$) = $2.70{\text{g/cm}}^3=2700{\text{kg/m}}^3$
• Molar mass ($M$) = $26.98\text{g/mol}$
• Avogadro’s number ($N_A$) = $6.022\times 10^{23}{\text{mol}}^{-1}$

Steps:

1. Calculate electron density ($n$):

   • Aluminum has 3 valence electrons/atom.
   • Number density of atoms: $$n_{\text{atoms}}=\frac{\rho N_A}{M}=\frac{(2700)(6.022\times 10^{23})}{26.98\times 10^{-3}}=6.02\times 10^{28}{\text{m}}^{-3}$$
   • Electron density: $$n=3\times n_{\text{atoms}}=1.806\times 10^{29}{\text{m}}^{-3}$$

2. Calculate $E_F$: $$E_F=\frac{{\hslash }^2}{2m}{\left(3{\pi }^{2}n\right)}^{2/3}=\frac{(1.05\times 10^{-34}{)}^2}{2(9.11\times 10^{-31})}{\left(3{\pi }^2\times 1.806\times 10^{29}\right)}^{2/3}$$ $$E_F=11.6\text{eV}$$ Close to the given $12\text{eV}$ (discrepancy due to effective mass).

3. Effective Mass ($m^{\ast }$): Given experimental $E_F=12\text{eV}$, solve for $m^{\ast }$: $$12=\frac{{\hslash }^2}{2m^{\ast }}{\left(3{\pi }^{2}n\right)}^{2/3}$$ $$m^{\ast }=\frac{{\hslash }^2}{2E_F}{\left(3{\pi }^{2}n\right)}^{2/3}=1.06m_e$$ Aluminum’s electrons act as if they’re 6% heavier than free electrons.

5. Torque on a Current Loop in a Magnetic Field

Derivation

A current loop in a magnetic field experiences:

1. Force on each segment: $d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}$
2. Net torque: $\tau =\mu \times \mathbf{B}$, where $\mu =IA$ (dipole moment).

For a rectangular loop (side lengths $a,b$) in a uniform field: $$\tau =NIAB\sin \theta$$

• $N$: Number of turns
• $\theta$: Angle between $\mu$ and $B$

Real-World Example: Electric Meters (Nepal Electricity Authority)

• How it works: A coil rotates in Earth’s magnetic field, with torque balanced by a spring.
• Equation: $${\tau }_{\text{magnetic}}={\tau }_{\text{spring}}⟹NIAB\sin \theta =k\theta$$
  • Measures current via angular deflection ($\theta$).

6. Experimental Techniques

Hall Effect Setup

1. Components:

   • Current source ($I$)
   • Magnetic field ($B$, via Helmholtz coils)
   • Voltage probes (across width)
   • Nanovoltmeter (to measure $V_H$)

2. Procedure:

   • Vary $I$ or $B$ and record $V_H$.
   • Plot $V_H$ vs. $I$ (should be linear).

Fermi Energy Measurement

1. Photoelectric Effect:

   • Shine light of varying frequency ($\nu$) on metal.
   • Measure stopping potential ($V_0$).
   • Plot $V_0$ vs. $\nu$: slope = $h/e$, intercept = $-E_F/e$.

2. Example Data (for aluminum):

   $\nu \times 10^{14}\text{Hz}$	$V_0(\text{V})$
   5.0	0.5
   6.0	1.7
   7.0	2.9

   • Slope = $h/e=4.14\times 10^{-15}\text{V}\cdot \text{s}$
   • Intercept = $-E_F/e=-11.6\text{V}$ → $E_F=11.6\text{eV}$

## In the Real World

1. eSewa’s Payment Terminals

   • What it uses: Hall-effect sensors to measure current in payment verification circuits.
   • Why? Ensures fraud detection (e.g., if current spikes, it flags a fake card).

2. Ncell’s Antenna Design

   • What it uses: Torque on current loops ($\tau =NIAB$) to optimize antenna alignment with Earth’s magnetic field.
   • Why? Reduces signal loss during monsoon storms (when $B$ fluctuates).

3. Daraz’s Warehouse Automation

   • What it uses: Hall sensors in conveyor belts to track package movement (non-contact, no wear).
   • How? A magnet on packages triggers a Hall switch → updates inventory in real time.

4. NEPSE’s Stock Exchange Sensors

   • What it uses: Hall probes to monitor electromagnetic interference near trading servers.
   • Why? Prevents data corruption during high-frequency trades.

5. Khalti’s NFC Payments

   • What it uses: Magnetic dipole moments in NFC chips (like a tiny current loop).
   • How? When you tap, the chip’s $\mu$ induces a current in the reader → authenticates payment.

## Exam Tip

1. Hall Effect Questions:

   • Always draw a diagram showing $I$, $B$, and $V_H$ directions.
   • Remember: Electrons move opposite to conventional current → $V_H$ flips sign!
   • For semiconductors, state whether $R_H$ is positive (p-type) or negative (n-type).

2. Magnetic Dipole Moment:

   • Atomic $\mu$: Use $\mu ={\mu }_B\sqrt{j(j+1)}$.
   • Current loop $\mu$: $\mu =IA$.
   • Torque: $\tau =\mu B\sin \theta$ (always ask for maximum torque scenario: $\theta =90^{\circ }$).

3. Fermi Energy Calculations:

   • Step 1: Calculate electron density ($n$) from density and molar mass.
   • Step 2: Plug into $E_F$ formula (or use photoelectric data).
   • Effective mass: If $E_F$ is given experimentally, solve for $m^{\ast }$ using the same formula.

4. Torque on Loops:

   • Key formula: $\tau =NIAB\sin \theta$.
   • Trick: For a square loop, $A=a^2$; for a circular loop, $A=\pi r^2$.
   • Real-world link: Always relate to electric meters or motor design.

5. Units and Signs:

   • $B$ is in tesla (T), $I$ in amperes (A), $V_H$ in volts (V).
   • Hall coefficient ($R_H$) is in m³/C (negative for electrons, positive for holes).

## Common Mistakes to Avoid

• Assuming $V_H$ is large: It’s usually millivolts—use sensitive meters!
• Ignoring carrier type: Copper (metal) has electrons; silicon (semiconductor) can have holes.
• Forgetting $\sin \theta$: Torque is zero when $\theta =0^{\circ }$ or $180^{\circ }$.
• Mixing $E_F$ formulas: For free electrons, use the density-based formula; for experimental data, use photoelectric plots.