understand the origin of the Hall voltage and derive and use the expression VH = BI / (ntq), where t = thickness

20.1 Concept of a Magnetic Field

Definition

• The magnetic field at a point is defined by the force that would act on a unit‑length straight conductor carrying a current of 1 A, placed perpendicular to the field.
• Mathematically,

  \[

  \boxed{B=\frac{F}{I\,L\sin\theta}\qquad\bigl[\text{T (tesla)}\bigr]}

  \]

  where \(F\) is the magnetic force, \(L\) the length of the conductor and \(\theta\) the angle between the direction of the conductor and the magnetic field.

Field‑line representation

• Field lines are drawn so that the tangent at any point gives the direction of \(\mathbf B\); the density of lines indicates the magnitude.
• Examples:

  • Bar magnet – lines emerge from the north pole and enter the south pole.
  • Current‑carrying straight wire – concentric circles around the wire (right‑hand rule).

Link to the Lorentz force

The force on a moving charge \(q\) with velocity \(\mathbf v\) in a magnetic field is

\[

\boxed{\mathbf F = q\,\mathbf v \times \mathbf B}

\]

This is the microscopic origin of the macroscopic force on a current‑carrying conductor (section 20.2).

20.2 Force on a Current‑Carrying Conductor (Cambridge 20.2)

2.1 General formula

\[

\boxed{\mathbf F = I\,\mathbf L \times \mathbf B \qquad\text{or}\qquad

F = B I L \sin\theta}

\]

• \(\mathbf L\) is a vector of magnitude \(L\) directed along the conventional current (from + to –).
• \(\theta\) is the angle between \(\mathbf L\) and \(\mathbf B\); the \(\sin\theta\) factor accounts for the component of \(\mathbf B\) perpendicular to the wire.
• Direction of \(\mathbf F\) is given by Fleming’s left‑hand rule (first finger = \(\mathbf B\), second finger = \(I\), thumb = \(\mathbf F\)).
• Units check: \([F]=\text{N},\;[I]=\text{A},\;[L]=\text{m},\;[B]=\text{T}\;\Rightarrow\; \text{A·m·T} = \text{N}\).

2.2 Worked example – non‑perpendicular field

A straight wire of length \(L=0.15\;\text{m}\) carries a current \(I=3.0\;\text{A}\). It is placed in a uniform magnetic field \(B=0.40\;\text{T}\) making an angle \(\theta =30^{\circ}\) with the wire. Find the magnitude of the force.

1. Use the scalar form \(F = B I L \sin\theta\).
2. \(\sin30^{\circ}=0.5\).
3. \(F = 0.40 \times 3.0 \times 0.15 \times 0.5 = 9.0\times10^{-2}\;\text{N}\).

2.3 Curved conductors – integration

For a wire of arbitrary shape the differential force is

\[

\boxed{d\mathbf F = I\,d\mathbf l \times \mathbf B}

\]

and the total force is obtained by integration:

\[

\mathbf F = I\int_{\text{wire}} d\mathbf l \times \mathbf B .

\]

Example (semicircular wire): A semicircle of radius \(r\) lies in a uniform field \(\mathbf B\) directed into the page. The current flows clockwise. The magnitude of the force is

\[

F = I B \int_{0}^{\pi} r\,d\phi = I B r \pi .

\]

20.3 Hall Effect and Hall Voltage (Cambridge 20.3)

3.1 What is the Hall effect?

• When a current‑carrying conductor is placed in a magnetic field, the magnetic part of the Lorentz force pushes the moving charge carriers toward one side of the material.
• This charge separation creates a transverse electric field \(\mathbf E_H\) (the Hall field) that grows until it balances the magnetic force.
• At equilibrium:

  \[

  q\,\mathbf EH = q\,\mathbf vd \times \mathbf B

  \;\Longrightarrow\;

  \boxed{\mathbf EH = \mathbf vd \times \mathbf B } .

  \]

• The potential difference measured across the thickness \(t\) of the plate is the Hall voltage \(V_H\).

3.2 Geometry used in the derivation

Consider a rectangular plate of:

• width \(w\) (direction of the current),
• thickness \(t\) (direction of the Hall voltage),
• length \(L\) (along the magnetic field, not needed in the final formula).

Current flows along the length of the plate, magnetic field \(\mathbf B\) is perpendicular to both \(\mathbf I\) and the plate surface, and the Hall voltage is measured across the thickness \(t\).

3.3 Derivation of the Hall‑voltage expression

1. Current density: \(\displaystyle \mathbf J = n q \mathbf v_d\). For a uniform current,

   \[

   I = J A = n q v_d (w t) .

   \]

2. Solve for the drift speed:

   \[

   v_d = \frac{I}{n q w t}.

   \]

3. Magnitude of the Hall field (since \(\mathbf v_d \perp \mathbf B\)):

   \[

   EH = vd B = \frac{I B}{n q w t}.

   \]

4. The Hall voltage is the field multiplied by the distance over which it acts – the thickness \(t\):

   \[

   \boxed{VH = EH \, t = \frac{B I}{n\,t\,q}} .

   \]

3.4 Hall coefficient

\[

\boxed{RH = \frac{EH}{J B} = \frac{1}{n q}}

\]

• Units: \(\text{m}^3\text{C}^{-1}\) (equivalently \(\Omega\!\cdot\!\text{T}^{-1}\)).
• The sign of \(R_H\) tells the nature of the dominant carriers:

  • \(R_H>0\) → positive carriers (holes).
  • \(R_H<0\) → negative carriers (electrons).

3.5 Assumptions & experimental realities

• Uniform current density across the cross‑section.
• Homogeneous magnetic field over the whole sample.
• Negligible edge effects – the Hall contacts are placed well within the plate.
• Deviations (e.g., non‑uniform \(B\) or current crowding) lead to systematic errors in the measured \(V_H\).

3.6 Typical values (AO1 checklist)

3.7 Experimental set‑up (AO2)

• Apparatus: rectangular metal/semiconductor strip, DC power supply, ammeter, high‑sensitivity voltmeter, Helmholtz coils or permanent magnets (to give a uniform \(\mathbf B\)), gauss‑meter, non‑magnetic mounting jig, calipers for measuring \(w\) and \(t\).
• Procedure (outline):

  1. Measure the width \(w\) and thickness \(t\) of the strip with a caliper.
  2. Connect the strip in series with an ammeter and a variable DC supply; set a known current \(I\) and record it.
  3. Switch on the magnetic field; note its magnitude (using the gauss‑meter) and direction.
  4. Connect the voltmeter across the two Hall contacts (on opposite faces of thickness \(t\)). Record the Hall voltage \(V_H\) and its polarity.
  5. Repeat for several values of \(I\) (or \(B\)) to verify the linear relationship \(V_H \propto I B\).

• Safety:

  • Do not exceed the current rating of the sample – overheating can permanently damage the material.
  • Secure strong permanent magnets; keep ferromagnetic tools away to avoid sudden attraction.
  • Use insulated leads and keep the voltmeter probes away from the power‑supply terminals to prevent short‑circuits.
  • Handle the gauss‑meter probe carefully – the field sensor is fragile.

3.8 Worked examples

Example 1 – Determining carrier density

Given: copper strip, \(t = 2.0\times10^{-4}\,\text{m}\); \(I = 2.5\ \text{A}\); \(B = 0.5\ \text{T}\); measured \(V_H = 3.2\times10^{-5}\ \text{V}\). Find \(n\).

1. Re‑arrange the Hall‑voltage equation:

   \[

   n = \frac{B I}{t q V_H}.

   \]

2. Insert values (\(q = e = 1.60\times10^{-19}\ \text{C}\)):

   \[

   n = \frac{0.5 \times 2.5}{(2.0\times10^{-4})(1.60\times10^{-19})(3.2\times10^{-5})}

   = 1.2\times10^{27}\ \text{m}^{-3}.

   \]

3. Interpretation: The result is of the correct order of magnitude for a metal; the simple single‑carrier model under‑estimates the true electron density in copper (\(\approx 8.5\times10^{28}\ \text{m}^{-3}\)).

Example 2 – Determining the magnetic field

Given: a p‑type semiconductor plate, \(t = 5.0\times10^{-4}\,\text{m}\); current \(I = 1.0\ \text{A}\); carrier density \(n = 5.0\times10^{22}\ \text{m}^{-3}\); measured Hall voltage \(V_H = 8.0\times10^{-6}\ \text{V}\). Find the magnetic field \(B\).

1. From \(V_H = \dfrac{B I}{n t q}\) solve for \(B\):

   \[

   B = \frac{V_H n t q}{I}.

   \]

2. Insert numbers (\(q = +e\) for holes):

   \[

   B = \frac{8.0\times10^{-6}\times5.0\times10^{22}\times5.0\times10^{-4}\times1.60\times10^{-19}}{1.0}

   \approx 0.32\ \text{T}.

   \]

3. Thus a magnetic field of about \(0.32\ \text{T}\) is required to produce the observed Hall voltage.

3.9 Common pitfalls (AO3)

• Confusing thickness \(t\) (distance between Hall contacts) with width \(w\) (direction of current). The Hall voltage is always measured across \(t\).
• Ignoring the sign of \(q\); the polarity of \(V_H\) tells you whether the dominant carriers are electrons (\(-e\)) or holes (\(+e\)).
• Using magnetic field values in gauss without converting to tesla (1 T = 10⁴ G).
• Forgetting the \(\sin\theta\) factor when the field is not perpendicular to the conductor.
• Assuming a perfectly uniform field; in practice field gradients introduce systematic error.

20.4 Magnetic Fields Due to Currents

4.1 Biot–Savart law (qualitative)

Every small element \(d\mathbf l\) of a current‑carrying conductor produces a magnetic field at a point P given by

\[

d\mathbf B = \frac{\mu_0}{4\pi}\,\frac{I\,d\mathbf l \times \hat{\mathbf r}}{r^{2}},

\]

where \(\hat{\mathbf r}\) is the unit vector from the element to P and \(r\) is the distance.

4.2 Field of a long straight wire

\[

\boxed{B = \frac{\mu_0 I}{2\pi r}}

\qquad\bigl[\mu_0 = 4\pi\times10^{-7}\ \text{T·m·A}^{-1}\bigr]

\]

• Direction given by the right‑hand grip rule (thumb = current, fingers curl in the direction of \(\mathbf B\)).
• Field lines are concentric circles centred on the wire.

4.3 Field inside a long solenoid

\[

\boxed{B = \mu_0 n I}

\]

• \(n\) = number of turns per unit length (turns · m\(^{-1}\)).
• Field is uniform and parallel to the axis inside the solenoid, negligible outside.

4.4 Field of a toroid

\[

\boxed{B = \frac{\mu_0 N I}{2\pi r}}

\]

• \(N\) = total number of turns, \(r\) = mean radius of the toroid.
• Field is confined to the core; outside the toroid \(B\approx0\).

4.5 Quick reference table

Summary (AO1)

• Magnetic flux density \(B\) is defined via the force on a 1 A, 1 m conductor placed perpendicular to the field.
• Force on a conductor: \(\mathbf F = I\,\mathbf L \times \mathbf B\) (or \(F = B I L \sin\theta\)).
• For curved conductors use \(d\mathbf F = I\,d\mathbf l \times \mathbf B\) and integrate.
• Hall effect: magnetic force separates charge carriers, creating a transverse electric field that balances the force.
• Hall voltage for a plate of thickness \(t\):

  \[

  V_H = \frac{B I}{n t q}.

  \]

• Hall coefficient \(R_H = 1/(n q)\); its sign identifies the dominant carrier type.
• Magnetic fields produced by currents are described by the Biot–Savart law; useful results are the straight‑wire, solenoid and toroid formulas.
• Experimental measurement of \(V_H\) provides a direct method for determining carrier density, sign, and for checking the uniformity of the magnetic field.