#set document(title: "12.1 The Biot-Savart Law", 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")) == 12.1#h(0.6em)The Biot-Savart Law We have seen that mass produces a gravitational field and also interacts with that field. Charge produces an electric field and also interacts with that field. Since moving charge (that is, current) interacts with a magnetic field, we might expect that it also creates that field—and it does. The equation used to calculate the magnetic field produced by a current is known as the Biot-Savart law. It is an empirical law named in honor of two scientists who investigated the interaction between a straight, current-carrying wire and a permanent magnet. This law enables us to calculate the magnitude and direction of the magnetic field produced by a current in a wire. The #strong[Biot-Savart law] states that at any point #emph[P] , the magnetic field #math.equation(block: false, alt: "d B →")[$d arrow(B)$] due to an element #math.equation(block: false, alt: "d l →")[$d arrow(l)$] of a current-carrying wire is given by #math.equation(block: true, alt: "d B → equals the fraction μ sub 0 over 4 π the fraction I d l → times r ^ over r squared .")[$d arrow(B) = frac(μ_(0), 4 π) #h(0.2em) frac(I d arrow(l) #h(0.2em) × #h(0.2em) hat(r), r^(2)) .$] #figure(figph[This figure demonstrates Biot-Savart Law. A current dI flows through a magnetic wire. A point P is located at the distance r from the wire. A vector r to the point P forms an angle theta with the wire. Magnetic field dB exists in the point P.], alt: "This figure demonstrates Biot-Savart Law. A current dI flows through a magnetic wire. A point P is located at the distance r from the wire. A vector r to the point P forms an angle theta with the wire. Magnetic field dB exists in the point P.", caption: [A current element #math.equation(block: false, alt: "I d l →")[$I d arrow(l)$] produces a magnetic field at point #emph[P] given by the Biot-Savart law.]) The constant #math.equation(block: false, alt: "μ sub 0")[$μ_(0)$] is known as the #strong[permeability of free space] and is exactly #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "μ sub 0 equals 4 π times 10 to the power −7 T times m/A")[$μ_(0) = 4 π #h(0.2em) × #h(0.2em) 10^("−7") "T" ⋅ "m/A"$] ] in the SI system. The infinitesimal wire segment #math.equation(block: false, alt: "d l →")[$d arrow(l)$] is in the same direction as the current #emph[I] (assumed positive), #emph[r] is the distance from #math.equation(block: false, alt: "d l →")[$d arrow(l)$] to #emph[P] and #math.equation(block: false, alt: "r ^")[$hat(r)$] is a unit vector that points from #math.equation(block: false, alt: "d l →")[$d arrow(l)$] to #emph[P], as shown in the figure. The direction of #math.equation(block: false, alt: "d B →")[$d arrow(B)$] is determined by applying the right-hand rule to the vector product #math.equation(block: false, alt: "d l → times r ^ .")[$d arrow(l) #h(0.2em) × #h(0.2em) hat(r) .$] The magnitude of #math.equation(block: false, alt: "d B →")[$d arrow(B)$] is #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "d B equals the fraction μ sub 0 over 4 π the fraction I d l sin θ over r squared")[$d B = frac(μ_(0), 4 π) #h(0.2em) frac(I #h(0.2em) d l #h(0.2em) sin #h(0.1em) θ, r^(2))$] ] where #math.equation(block: false, alt: "θ")[$θ$] is the angle between #math.equation(block: false, alt: "d l →")[$d arrow(l)$] and #math.equation(block: false, alt: "r ^ .")[$hat(r) .$] Notice that if #math.equation(block: false, alt: "θ equals 0 ,")[$θ = 0 ,$] then #math.equation(block: false, alt: "d B → equals 0 → .")[$d arrow(B) = arrow(0) .$] The field produced by a current element #math.equation(block: false, alt: "I d l →")[$I d arrow(l)$] has no component parallel to #math.equation(block: false, alt: "d l → .")[$d arrow(l) .$] The magnetic field due to a finite length of current-carrying wire is found by integrating along the wire, giving us the usual form of the Biot-Savart law. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Biot-Savart law] The magnetic field #math.equation(block: false, alt: "B →")[$arrow(B)$] due to a current-carrying wire is found by #math.equation(block: true, alt: "B → equals the fraction μ sub 0 over 4 π ∫ wire the fraction I d l → times r ^ over r squared .")[$arrow(B) = frac(μ_(0), 4 π) display(limits(∫)_("wire") frac(I #h(0.2em) d arrow(l) #h(0.2em) × #h(0.2em) hat(r), r^(2))) .$] ] Since this is a vector integral, contributions from different current elements may not point in the same direction. Consequently, the integral is often difficult to evaluate, even for fairly simple geometries. The following strategy may be helpful. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solving Biot-Savart Problems] To solve Biot-Savart law problems, the following steps are helpful: + Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing #math.equation(block: false, alt: "B →")[$arrow(B)$] and #math.equation(block: false, alt: "d l → ,")[$d arrow(l) ,$] Ampère’s law may be the preferred method to solve the question, which will be discussed in Ampère’s Law. + Draw the current element length #math.equation(block: false, alt: "d l →")[$d arrow(l)$] and the unit vector #math.equation(block: false, alt: "r ^ ,")[$hat(r) ,$] noting that #math.equation(block: false, alt: "d l →")[$d arrow(l)$] points in the direction of the current and #math.equation(block: false, alt: "r ^")[$hat(r)$] points from the current element toward the point where the field is desired. + Calculate the cross product #math.equation(block: false, alt: "d l → times r ^ .")[$d arrow(l) #h(0.2em) × #h(0.2em) hat(r) .$] The resultant vector gives the direction of the magnetic field according to the Biot-Savart law. + Use and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration. + Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part. ] #examplebox("Example 1")[Calculating Magnetic Fields of Short Current Segments][ A short wire of length 1.0 cm carries a current of 2.0 A in the vertical direction . The rest of the wire is shielded so it does not add to the magnetic field produced by the wire. Calculate the magnetic field at point #emph[P], which is 1 meter from the wire in the #emph[x]-direction. #figure(figph[This figure shows a wire I with a short unshielded piece dI that carries current. Point P is located at the distance x from the wire. A vector to the point P from dI forms an angle theta with the wire. The length of the vector is the square root of the sums of squares of x and I.], alt: "This figure shows a wire I with a short unshielded piece dI that carries current. Point P is located at the distance x from the wire. A vector to the point P from dI forms an angle theta with the wire. The length of the vector is the square root of the sums of squares of x and I.", caption: [A small line segment carries a current #emph[I] in the vertical direction. What is the magnetic field at a distance #emph[x] from the segment?]) Strategy We can determine the magnetic field at point #emph[P] using the Biot-Savart law. Since the current segment is much smaller than the distance #emph[x], we can drop the integral from the expression. The integration is converted back into a summation, but only for small #emph[dl], which we now write as #math.equation(block: false, alt: "Δ l .")[$"Δ" l .$] Another way to think about it is that each of the radius values is nearly the same, no matter where the current element is on the line segment, if #math.equation(block: false, alt: "Δ l")[$"Δ" l$] is small compared to #emph[x]. The angle #math.equation(block: false, alt: "θ")[$θ$] is calculated using a tangent function. Using the numbers given, we can calculate the magnetic field at #emph[P]. Solution The angle between #math.equation(block: false, alt: "Δ l →")[$"Δ" arrow(l)$] and #math.equation(block: false, alt: "r ^")[$hat(r)$] is calculated from trigonometry, knowing the distances #emph[l] and #emph[x] from the problem: #math.equation(block: true, alt: "θ equals tan to the power −1 open parenthesis the fraction 1 m over 0.01 m close parenthesis equals 89.4 ° .")[$θ = "tan"^(−1) ( frac(1 #h(0.2em) "m", 0.01 #h(0.2em) "m") ) = 89.4 "°" .$] The magnetic field at point #emph[P] is calculated by the Biot-Savart law: #math.equation(block: true, alt: "B equals the fraction μ sub 0 over 4 π the fraction I Δ l sin θ over r squared equals open parenthesis 1 times 10 to the power −7 T times m/A close parenthesis open parenthesis the fraction 2 A open parenthesis 0.01 m close parenthesis sin open parenthesis 89.4 ° close parenthesis over open parenthesis 1 m close parenthesis squared close parenthesis equals 2.0 times 10 to the power −9 T .")[$B = frac(μ_(0), 4 π) #h(0.2em) frac(I "Δ" l #h(0.2em) "sin" #h(0.1em) θ, r^(2)) = ( 1 #h(0.2em) × #h(0.2em) 10^(−7) "T" ⋅ "m/A" ) ( frac(2 #h(0.2em) "A" ( 0.01 #h(0.2em) "m" ) #h(0.1em) "sin" ( 89.4 "°" ), attach(( 1 #h(0.2em) "m" ), t: 2)) ) = 2.0 #h(0.2em) × #h(0.2em) 10^(−9) "T" .$] From the right-hand rule and the Biot-Savart law, the field is directed into the page. Significance This approximation is only good if the length of the line segment is very small compared to the distance from the current element to the point. If not, the integral form of the Biot-Savart law must be used over the entire line segment to calculate the magnetic field. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Using , at what distance would #emph[P] have to be to measure a magnetic field half of the given answer? #solutionbox[ 1.41 meters ] ] #examplebox("Example 2")[Calculating Magnetic Field of a Circular Arc of Wire][ A wire carries a current #emph[I] in a circular arc with radius #emph[R] swept through an arbitrary angle #math.equation(block: false, alt: "θ")[$θ$] . Calculate the magnetic field at the center of this arc at point #emph[P]. #figure(figph[This figure shows a piece of wire in the shape of a circular arc with radius R swept through an arbitrary angle theta. Wire carries a current dI. Point P is located at the center. A vector r to the point P is perpendicular to the vector dI.], alt: "This figure shows a piece of wire in the shape of a circular arc with radius R swept through an arbitrary angle theta. Wire carries a current dI. Point P is located at the center. A vector r to the point P is perpendicular to the vector dI.", caption: [A wire segment carrying a current #emph[I]. The path #math.equation(block: false, alt: "d l →")[$d arrow(l)$] and radial direction #math.equation(block: false, alt: "r ^")[$hat(r)$] are indicated.]) Strategy We can determine the magnetic field at point #emph[P] using the Biot-Savart law. The radial and path length directions are always at a right angle, so the cross product turns into multiplication. We also know that the distance along the path #emph[dl] is related to the radius times the angle #math.equation(block: false, alt: "θ")[$θ$] (in radians). Then we can pull all constants out of the integration and solve for the magnetic field. Solution The Biot-Savart law starts with the following equation: #math.equation(block: true, alt: "B → equals the fraction μ sub 0 over 4 π ∫ wire the fraction I d l → times r ^ over r squared .")[$arrow(B) = frac(μ_(0), 4 π) display(limits(∫)_("wire") frac(I d arrow(l) #h(0.2em) × #h(0.2em) hat(r), r^(2))) .$] As we integrate along the arc, all the contributions to the magnetic field are in the same direction (out of the page), so we can work with the magnitude of the field. The cross product turns into multiplication because the path #emph[dl] and the radial direction are perpendicular. We can also substitute the arc length formula, #math.equation(block: false, alt: "d l equals r d θ")[$d l = r d θ$]: #math.equation(block: true, alt: "B equals the fraction μ sub 0 over 4 π ∫ wire the fraction I r d θ over r squared .")[$B = frac(μ_(0), 4 π) display(limits(∫)_("wire") frac(I r #h(0.2em) d θ, r^(2))) .$] The current and radius can be pulled out of the integral because they are the same regardless of where we are on the path. This leaves only the integral over the angle, #math.equation(block: true, alt: "B equals the fraction μ sub 0 I over 4 π r ∫ wire d θ .")[$B = frac(μ_(0) I, 4 π r) display(limits(∫)_("wire") d θ) .$] The angle varies on the wire from 0 to #math.equation(block: false, alt: "θ")[$θ$]; hence, the result is #math.equation(block: true, alt: "B equals the fraction μ sub 0 I θ over 4 π r .")[$B = frac(μ_(0) I θ, 4 π r) .$] Significance The direction of the magnetic field at point #emph[P] is determined by the right-hand rule, as shown in the previous chapter. If there are other wires in the diagram along with the arc, and you are asked to find the net magnetic field, find each contribution from a wire or arc and add the results by superposition of vectors. Make sure to pay attention to the direction of each contribution. Also note that in a symmetric situation, like a straight or circular wire, contributions from opposite sides of point #emph[P] cancel each other. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The wire loop forms a full circle of radius #emph[R] and current #emph[I]. What is the magnitude of the magnetic field at the center? #solutionbox[ #math.equation(block: true, alt: "the fraction μ sub 0 I over 2 R")[$frac(μ_(0) I, 2 R)$] ] ] === Summary - The magnetic field created by a current-carrying wire is found by the Biot-Savart law. - The current element #math.equation(block: false, alt: "I d l →")[$I d arrow(l)$] produces a magnetic field a distance #emph[r] away. === Conceptual Questions For calculating magnetic fields, what are the advantages and disadvantages of the Biot-Savart law? #solutionbox[ Biot-Savart law’s advantage is that it works with any magnetic field produced by a current loop. The disadvantage is that it can take a long time. ] Describe the magnetic field due to the current in two wires connected to the two terminals of a source of emf and twisted tightly around each other. How can you decide if a wire is infinite? #solutionbox[ If you were to go to the start of a line segment and calculate the angle #math.equation(block: false, alt: "θ")[$θ$] to be approximately #math.equation(block: false, alt: "0 °")[$0 "°"$], the wire can be considered infinite. This judgment is based also on the precision you need in the result. ] Identical currents are carried in two circular loops; however, one loop has twice the diameter as the other loop. Compare the magnetic fields created by the loops at the center of each loop. === Problems A 10-A current flows through the wire shown. What is the magnitude of the magnetic field due to a 0.5-mm segment of wire as measured at (a) point A and (b) point B? #figure(figph[This figure shows a piece of wire. Point A is located 3 centimeters above the 0.5 mm segment of wire. Point B is located 4 centimeters to the right of point A.], alt: "This figure shows a piece of wire. Point A is located 3 centimeters above the 0.5 mm segment of wire. Point B is located 4 centimeters to the right of point A.", caption: none) Ten amps flow through a square loop where each side is 20 cm in length. At each corner of the loop is a 0.01-cm segment that connects the longer wires as shown. Calculate the magnitude of the magnetic field at the center of the loop. #figure(figph[A square loop is shown with rounded corners. There are no markings.], alt: "A square loop is shown with rounded corners. There are no markings.", caption: none) #solutionbox[ #math.equation(block: true, alt: "5.66 times 10 to the power −5 T")[$5.66 #h(0.2em) × #h(0.2em) 10^(−5) "T"$] ] What is the magnetic field at P due to the current #emph[I] in the wire shown? #figure(figph[This figure shows a current loop consisting of two concentric circular arcs and two parallel radial lines. Outer arc is located at the distance b from the center; inner arc is located at the distance a from the center.], alt: "This figure shows a current loop consisting of two concentric circular arcs and two parallel radial lines. Outer arc is located at the distance b from the center; inner arc is located at the distance a from the center.", caption: none) The accompanying figure shows a current loop consisting of two concentric circular arcs and two perpendicular radial lines. Determine the magnetic field at point P. #figure(figph[This figure shows a current loop consisting of two concentric circular arcs and two perpendicular radial lines. Outer arc is located at the distance b from the center; inner arc is located at the distance a from the center.], alt: "This figure shows a current loop consisting of two concentric circular arcs and two perpendicular radial lines. Outer arc is located at the distance b from the center; inner arc is located at the distance a from the center.", caption: none) #solutionbox[ #math.equation(block: false, alt: "B equals the fraction μ sub o I over 8 open parenthesis the fraction 1 over a minus the fraction 1 over b close parenthesis")[$B = frac(μ_(o) I, 8) ( frac(1, a) − #h(0.05em) frac(1, b) )$] out of the page ] Find the magnetic field at the center C of the rectangular loop of wire shown in the accompanying figure. #figure(figph[This figure shows a rectangular current loop. The length of the short side is b; the length of the long side is a. Point C is a center of the loop.], alt: "This figure shows a rectangular current loop. The length of the short side is b; the length of the long side is a. Point C is a center of the loop.", caption: none) Two long wires, one of which has a semicircular bend of radius #emph[R], are positioned as shown in the accompanying figure. If both wires carry a current #emph[I], how far apart must their parallel sections be so that the net magnetic field at P is zero? Does the current in the straight wire flow up or down? #figure(figph[This figure shows two parallel long wires located at a distance a from each other. One of the wires has a semicircular bend of radius R.], alt: "This figure shows two parallel long wires located at a distance a from each other. One of the wires has a semicircular bend of radius R.", caption: none) #solutionbox[ #math.equation(block: false, alt: "a equals the fraction 2 R over π")[$a = frac(2 R, π)$]; the current in the wire to the right must flow up the page. ]