#set document(title: "11.6 The Hall Effect", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


  
== 11.6#h(0.6em)The Hall Effect


  In 1879, E.H. #strong[Hall] devised an experiment that can be used to identify the sign of the predominant charge carriers in a conducting material. From a historical perspective, this experiment was the first to demonstrate that the charge carriers in most metals are negative.


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Visit this #link("https://openstax.org/l/21halleffect")[website] to find more information about the Hall effect.
]


  We investigate the #strong[Hall effect] by studying the motion of the free electrons along a metallic strip of width #emph[l] in a constant magnetic field . The electrons are moving from left to right, so the magnetic force they experience pushes them to the bottom edge of the strip. This leaves an excess of positive charge at the top edge of the strip, resulting in an electric field #emph[E] directed from top to bottom. The charge concentration at both edges builds up until the electric force on the electrons in one direction is balanced by the magnetic force on them in the opposite direction. Equilibrium is reached when:


  #math.equation(block: true, alt: "e E equals e v sub d B")[$e E = e v_(d) B$]
  where #emph[e] is the magnitude of the electron charge, #math.equation(block: false, alt: "v sub d")[$v_(d)$] is the drift speed of the electrons, and #emph[E] is the magnitude of the electric field created by the separated charge. Solving this for the drift speed results in


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "v sub d equals the fraction E over B .")[$v_(d) = frac(E, B) .$]
]


  #figure(figph[An illustration of the Hall effect: In both figures, the current in the strip is to the left and the magnetic field points into the page. In figure a, a negative charge is moving to the right with velocity v d. Positive charges accumulate at the top of the strip, negative charges at the bottom of the strip. An electric field E sub H points down. The moving charge experiences an upward force e E sub H and a downward force e v sub d B. In figure b, a positive charge is moving to the left with velocity v d. Negative charges accumulate at the top of the strip, positive charges at the bottom of the strip. An electric field E sub H points up. The moving charge experiences an upward force e E sub H and a downward force e v sub d B.], alt: "An illustration of the Hall effect: In both figures, the current in the strip is to the left and the magnetic field points into the page. In figure a, a negative charge is moving to the right with velocity v d. Positive charges accumulate at the top of the strip, negative charges at the bottom of the strip. An electric field E sub H points down. The moving charge experiences an upward force e E sub H and a downward force e v sub d B. In figure b, a positive charge is moving to the left with velocity v d. Negative charges accumulate at the top of the strip, positive charges at the bottom of the strip. An electric field E sub H points up. The moving charge experiences an upward force e E sub H and a downward force e v sub d B.", caption: [In the Hall effect, a potential difference between the top and bottom edges of the metal strip is produced when moving charge carriers are deflected by the magnetic field. (a) Hall effect for negative charge carriers; (b) Hall effect for positive charge carriers.])


  A scenario where the electric and magnetic fields are perpendicular to one another is called a crossed-field situation. If these fields produce equal and opposite forces on a charged particle with the velocity that equates the forces, these particles are able to pass through an apparatus, called a #strong[velocity selector], undeflected. This velocity is represented in . Any other velocity of a charged particle sent into the same fields would be deflected by the magnetic force or electric force.


  Going back to the Hall effect, if the current in the strip is #emph[I], then from Current and Resistance, we know that


  #math.equation(block: true, alt: "I equals n e v sub d A")[$I = n e v_(d) A$]
  where #emph[n] is the number of charge carriers per volume and #emph[A] is the cross-sectional area of the strip. Combining the equations for #math.equation(block: false, alt: "v sub d")[$v_(d)$] and #emph[I] results in


  #math.equation(block: true, alt: "I equals n e open parenthesis the fraction E over B close parenthesis A .")[$I = n e ( frac(E, B) ) A .$]
  The field #emph[E] is related to the potential difference #emph[V] between the edges of the strip by


  #math.equation(block: true, alt: "E equals the fraction V over l .")[$E = frac(V, l) .$]
  The quantity #emph[V] is called the Hall potential and can be measured with a voltmeter. Finally, combining the equations for #emph[I] and #emph[E] gives us


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "V equals the fraction I B l over n e A")[$V = frac(I B l, n e A)$]
]


  where the upper edge of the strip in  is positive with respect to the lower edge.


  We can also combine  and  to get an expression for the Hall voltage in terms of the magnetic field:


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "V equals B l v sub d .")[$V = B l v_(d) .$]
]


  What if the charge carriers are positive, as in ? For the same current #emph[I], the magnitude of #emph[V] is still given by . However, the upper edge is now negative with respect to the lower edge. Therefore, by simply measuring the sign of #emph[V], we can determine the sign of the majority charge carriers in a metal.


  Hall potential measurements show that electrons are the dominant charge carriers in most metals. However, Hall potentials indicate that for a few metals, such as tungsten, beryllium, and many semiconductors, the majority of charge carriers are positive. It turns out that conduction by positive charge is caused by the migration of missing electron sites (called holes) on ions. Conduction by holes is studied later in Condensed Matter Physics.


  The Hall effect can be used to measure magnetic fields. If a material with a known density of charge carriers #emph[n] is placed in a magnetic field and #emph[V] is measured, then the field can be determined from . In research laboratories where the fields of electromagnets used for precise measurements have to be extremely steady, a “Hall probe” is commonly used as part of an electronic circuit that regulates the field.


  #examplebox("Example 1")[Velocity Selector][
An electron beam enters a crossed-field velocity selector with magnetic and electric fields of 2.0 mT and #math.equation(block: false, alt: "6.0 times 10 cubed N/C,")[$6.0 #h(0.2em) × #h(0.2em) 10^(3) #h(0.2em) "N/C,"$] respectively. (a) What must the velocity of the electron beam be to traverse the crossed fields undeflected? If the electric field is turned off, (b) what is the acceleration of the electron beam and (c) what is the radius of the circular motion that results?


    Strategy
The electron beam is not deflected by either of the magnetic or electric fields if these forces are balanced. Based on these balanced forces, we calculate the velocity of the beam. Without the electric field, only the magnetic force is used in Newton’s second law to find the acceleration. Lastly, the radius of the path is based on the resulting circular motion from the magnetic force.


    Solution


    + The velocity of the unperturbed beam of electrons with crossed fields is calculated by :
#math.equation(block: true, alt: "v sub d equals the fraction E over B equals the fraction 6 times 10 cubed N / C over 2 times 10 to the power −3 T equals 3 times 10 to the power 6 m / s.")[$v_(d) = frac(E, B) = frac(6 #h(0.2em) × #h(0.2em) 10^(3) #h(0.2em) "N" "/" "C", 2 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "T") = 3 #h(0.2em) × #h(0.2em) 10^(6) "m" "/" "s."$]
+ The acceleration is calculated from the net force from the magnetic field, equal to mass times acceleration. The magnitude of the acceleration is:
#math.equation(block: true, alt: "m a, equals, q v B; a, equals, the fraction q v B over m equals the fraction open parenthesis 1.6 times 10 to the power −19 C close parenthesis open parenthesis 3 times 10 to the power 6 m/s close parenthesis open parenthesis 2 times 10 to the power −3 T close parenthesis over 9.1 times 10 to the power −31 kg equals 1.1 times 10 to the power 15 m/s squared .")[$m a & = & q v B \ a & = & frac(q v B, m) = frac(( 1.6 #h(0.2em) × #h(0.2em) 10^(−19) "C" ) ( 3 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "m/s" ) ( 2 #h(0.2em) × #h(0.2em) 10^(−3) "T" ), 9.1 #h(0.2em) × #h(0.2em) 10^(−31) "kg") = 1.1 #h(0.2em) × #h(0.2em) 10^(15) #h(0.2em) "m/s"^(2) .$]
+ The radius of the path comes from a balance of the circular and magnetic forces, or :
#math.equation(block: true, alt: "r equals the fraction m v over q B equals the fraction open parenthesis 9.1 times 10 to the power −31 kg close parenthesis open parenthesis 3 times 10 to the power 6 m/s close parenthesis over open parenthesis 1.6 times 10 to the power −19 C close parenthesis open parenthesis 2 times 10 to the power −3 T close parenthesis equals 8.5 times 10 to the power −3 m.")[$r = frac(m v, q B) = frac(( 9.1 #h(0.2em) × #h(0.2em) 10^(−31) "kg" ) ( 3 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "m/s" ), ( 1.6 #h(0.2em) × #h(0.2em) 10^(−19) "C" ) ( 2 #h(0.2em) × #h(0.2em) 10^(−3) "T" )) = 8.5 #h(0.2em) × #h(0.2em) 10^(−3) "m."$]


    Significance
If electrons in the beam had velocities above or below the answer in part (a), those electrons would have a stronger net force exerted by either the magnetic or electric field. Therefore, only those electrons at this specific velocity would make it through.
]


  #examplebox("Example 2")[The Hall Potential in a Silver Ribbon][
shows a silver ribbon whose cross section is 1.0 cm by 0.20 cm. The ribbon carries a current of 100 A from left to right, and it lies in a uniform magnetic field of magnitude 1.5 T. Using a density value of #math.equation(block: false, alt: "n equals 5.9 times 10 to the power 28")[$n = 5.9 #h(0.2em) × #h(0.2em) 10^(28)$] electrons per cubic meter for silver, find the Hall potential between the edges of the ribbon.


    #figure(figph[The silver ribbon is shown with current flowing to the right, a magnetic field pointing up, negative charges accumulating on the edge near us and positive charges accumulating on the far edge. The dimensions of the strip are 1.0 cm by 0.20 cm.], alt: "The silver ribbon is shown with current flowing to the right, a magnetic field pointing up, negative charges accumulating on the edge near us and positive charges accumulating on the far edge. The dimensions of the strip are 1.0 cm by 0.20 cm.", caption: [Finding the Hall potential in a silver ribbon in a magnetic field is shown.])


    Strategy
Since the majority of charge carriers are electrons, the polarity of the Hall voltage is that indicated in the figure. The value of the Hall voltage is calculated using :


    #math.equation(block: true, alt: "V equals the fraction I B l over n e A .")[$V = frac(I B l, n e A) .$]
    Solution
When calculating the Hall voltage, we need to know the current through the material, the magnetic field, the length, the number of charge carriers, and the area. Since all of these are given, the Hall voltage is calculated as:


    #math.equation(block: true, alt: "V equals the fraction I B l over n e A equals the fraction open parenthesis 100 A close parenthesis open parenthesis 1.5 T close parenthesis open parenthesis 1.0 times 10 to the power −2 m close parenthesis over open parenthesis 5.9 times 10 to the power 28 / m cubed close parenthesis open parenthesis 1.6 times 10 to the power −19 C close parenthesis open parenthesis 2.0 times 10 to the power −5 m squared close parenthesis equals 7.9 times 10 to the power −6 V.")[$V = frac(I B l, n e A) = frac(( 100 #h(0.2em) "A" ) ( 1.5 #h(0.2em) "T" ) ( 1.0 #h(0.2em) × #h(0.2em) 10^(−2) "m" ), ( 5.9 #h(0.2em) × #h(0.2em) 10^(28) "/" "m"^(3) ) ( 1.6 #h(0.2em) × #h(0.2em) 10^(−19) "C" ) ( 2.0 #h(0.2em) × #h(0.2em) 10^(−5) "m"^(2) )) = 7.9 #h(0.2em) × #h(0.2em) 10^(−6) "V."$]
    Significance
As in this example, the Hall potential is generally very small, and careful experimentation with sensitive equipment is required for its measurement.
]


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A Hall probe consists of a copper strip, #math.equation(block: false, alt: "n equals 8.5 times 10 to the power 28")[$n = 8.5 #h(0.2em) × #h(0.2em) 10^(28)$] electrons per cubic meter, which is 2.0 cm wide and 0.10 cm thick. What is the magnetic field when #emph[I] = 50 A and the Hall potential is (a) #math.equation(block: false, alt: "4.0 μ V")[$4.0 μ "V"$] and (b) #math.equation(block: false, alt: "6.0 μ V ?")[$6.0 μ "V" ?$]


    
    #solutionbox[
a. 1.1 T; b. 1.6 T
]
]


  
    
=== Summary


    - Perpendicular electric and magnetic fields exert equal and opposite forces for a specific velocity of entering particles, thereby acting as a velocity selector. The velocity that passes through undeflected is calculated by #math.equation(block: false, alt: "v equals the fraction E over B .")[$v = frac(E, B) .$]
- The Hall effect can be used to measure the sign of the majority of charge carriers for metals. It can also be used to measure a magnetic field.


  
  
    
=== Conceptual Questions


    
      Hall potentials are much larger for poor conductors than for good conductors with the same current. Why?


    
    #solutionbox[
Poor conductors have a lower charge carrier density, #emph[n], which, based on the Hall effect formula, relates to a higher Hall potential. Good conductors have a higher charge carrier density, thereby a lower Hall potential.
]


  
  
    
=== Problems


    
      A strip of copper is placed in a uniform magnetic field of magnitude 2.5 T. The Hall electric field is measured to be #math.equation(block: false, alt: "1.5 times 10 to the power −3 V/m .")[$1.5 #h(0.2em) × #h(0.2em) 10^(−3) "V/m" .$] (a) What is the drift speed of the conduction electrons? (b) Assuming that n = #math.equation(block: false, alt: "8.0 times 10 to the power 28")[$8.0 #h(0.2em) × #h(0.2em) 10^(28)$] electrons per cubic meter and that the cross-sectional area of the strip is #math.equation(block: false, alt: "5.0 times 10 to the power −6 m squared ,")[$5.0 #h(0.2em) × #h(0.2em) 10^(−6) "m"^(2) ,$] calculate the current in the strip. (c) What is the Hall coefficient 1/nq?


    
    
      The cross-sectional dimensions of the copper strip shown are 2.0 cm by 2.0 mm. The strip carries a current of 100 A, and it is placed in a magnetic field of magnitude #emph[B] = 1.5 T. What are the value and polarity of the Hall potential in the copper strip?


      #figure(figph[A horizontal 2.0 cm by 2.0 cm square copper strip has current I flowing through it to the right. A magnetic field, B, points up, perpendicular to the face of the strip.], alt: "A horizontal 2.0 cm by 2.0 cm square copper strip has current I flowing through it to the right. A magnetic field, B, points up, perpendicular to the face of the strip.", caption: none)


    
    #solutionbox[
#math.equation(block: true, alt: "5.8 times 10 to the power −6 V")[$5.8 #h(0.2em) × #h(0.2em) 10^(−6) "V"$]
]


    
      The magnitudes of the electric and magnetic fields in a velocity selector are #math.equation(block: false, alt: "1.8 times 10 to the power 5 V/m")[$1.8 #h(0.2em) × #h(0.2em) 10^(5) "V/m"$] and 0.080 T, respectively. (a) What speed must a proton have to pass through the selector? (b) Also calculate the speeds required for an alpha-particle and a singly ionized #math.equation(block: false, alt: "s O to the power 16")[$#h(0.5em) s O^(16)$] atom to pass through the selector.


    
    
      A charged particle moves through a velocity selector at constant velocity. In the selector, #emph[E] = #math.equation(block: false, alt: "1.0 times 10 to the power 4 N/C")[$1.0 #h(0.2em) × #h(0.2em) 10^(4) "N/C"$] and #emph[B] = 0.250 T. When the electric field is turned off, the charged particle travels in a circular path of radius 3.33 mm. Determine the charge-to-mass ratio of the particle.


    
    #solutionbox[
#math.equation(block: true, alt: "4.8 times 10 to the power 7 C/kg")[$4.8 #h(0.2em) × #h(0.2em) 10^(7) "C/kg"$]
]


    
      A Hall probe gives a reading of #math.equation(block: false, alt: "1.5 μ V")[$1.5 μ "V"$] for a current of 2 A when it is placed in a magnetic field of 1 T. What is the magnetic field in a region where the reading is #math.equation(block: false, alt: "2 μ V")[$2 μ "V"$] for 1.7 A of current?