#set document(title: "7.4 Determining Field from Potential", 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")) == 7.4#h(0.6em)Determining Field from Potential Recall that we were able, in certain systems, to calculate the potential by integrating over the electric field. As you may already suspect, this means that we may calculate the electric field by taking derivatives of the potential, although going from a scalar to a vector quantity introduces some interesting wrinkles. We frequently need #math.equation(block: false, alt: "E →")[$arrow("E")$] to calculate the force in a system; since it is often simpler to calculate the potential directly, there are systems in which it is useful to calculate #emph[V] and then derive #math.equation(block: false, alt: "E →")[$arrow("E")$] from it. In general, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of #math.equation(block: false, alt: "E →")[$arrow("E")$] and also in the direction of lower potential #emph[V]. Furthermore, the magnitude of #math.equation(block: false, alt: "E →")[$arrow("E")$] equals the rate of decrease of #emph[V] with distance. The faster #emph[V] decreases over distance, the greater the electric field. This gives us the following result. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Relationship between Voltage and Uniform Electric Field] In equation form, the relationship between voltage and uniform electric field is #math.equation(block: true, alt: "E equals minus the fraction Δ V over Δ s")[$E = − frac("Δ" V, "Δ" s)$] where #math.equation(block: false, alt: "Δ s")[$"Δ" s$] is the distance over which the change in potential #math.equation(block: false, alt: "Δ V")[$"Δ" V$] takes place. The minus sign tells us that #emph[E] points in the direction of decreasing potential. The electric field is said to be the gradient (as in grade or slope) of the electric potential. ] For continually changing potentials, #math.equation(block: false, alt: "Δ V")[$"Δ" V$] and #math.equation(block: false, alt: "Δ s")[$"Δ" s$] become infinitesimals, and we need differential calculus to determine the electric field. As shown in , if we treat the distance #math.equation(block: false, alt: "Δ s")[$"Δ" s$] as very small so that the electric field is essentially constant over it, we find that #math.equation(block: true, alt: "E sub s equals minus the fraction d V over d s .")[$E_(s) = − frac(d V, d s) .$] #figure(figph[The figure shows the electric field component of two points A and B separated by distance delta s and having a potential difference of delta V.], alt: "The figure shows the electric field component of two points A and B separated by distance delta s and having a potential difference of delta V.", caption: [The electric field component, #math.equation(block: false, alt: "E sub 1")[$E_(1)$], along the displacement #math.equation(block: false, alt: "Δ s")[$"Δ" s$] is given by #math.equation(block: false, alt: "E equals minus the fraction Δ V over Δ s")[$E = − frac("Δ" V, "Δ" s)$]. Note that #emph[A] and #emph[B] are assumed to be so close together that the field is constant along #math.equation(block: false, alt: "Δ s")[$"Δ" s$].]) Therefore, the electric field components in the Cartesian directions are given by #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "E sub x equals minus the fraction ∂ V over ∂ x , E sub y equals minus the fraction ∂ V over ∂ y , E sub z equals minus the fraction ∂ V over ∂ z .")[$E_(x) = − frac(∂ V, ∂ x) , #h(0.2em) E_(y) = − frac(∂ V, ∂ y) , #h(0.2em) E_(z) = − frac(∂ V, ∂ z) .$] ] This allows us to define the “grad” or “del” vector operator, which allows us to compute the gradient in one step. In Cartesian coordinates, it takes the form #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "∇ → equals i ^ the fraction ∂ over ∂ x plus j ^ the fraction ∂ over ∂ y plus k ^ the fraction ∂ over ∂ z .")[$arrow(∇) = hat("i") frac(∂, ∂ x) + hat("j") frac(∂, ∂ y) + hat("k") frac(∂, ∂ z) .$] ] With this notation, we can calculate the electric field from the potential with #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "E → equals − ∇ → V ,")[$arrow("E") = "−" arrow(∇) V ,$] ] a process we call calculating the #strong[gradient] of the potential. If we have a system with either cylindrical or spherical symmetry, we only need to use the del operator in the appropriate coordinates: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "Cylindrical: ∇ → equals r ^ the fraction ∂ over ∂ r plus φ ^ the fraction 1 over r the fraction ∂ over ∂ φ plus z ^ the fraction ∂ over ∂ z")[$"Cylindrical:" #h(0.2em) arrow(∇) = hat("r") frac(∂, ∂ r) + hat("φ") frac(1, r) #h(0.2em) frac(∂, ∂ φ) + hat("z") frac(∂, ∂ z)$] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "Spherical: ∇ → equals r ^ the fraction ∂ over ∂ r plus θ ^ the fraction 1 over r the fraction ∂ over ∂ θ plus φ ^ the fraction 1 over r sin θ the fraction ∂ over ∂ φ")[$"Spherical:" #h(0.2em) arrow(∇) = hat("r") frac(∂, ∂ r) + hat("θ") frac(1, r) #h(0.2em) frac(∂, ∂ θ) + hat("φ") frac(1, r #h(0.2em) "sin" #h(0.2em) θ) #h(0.2em) frac(∂, ∂ φ)$] ] #examplebox("Example 1")[Electric Field of a Point Charge][ Calculate the electric field of a point charge from the potential. Strategy The potential is known to be #math.equation(block: false, alt: "V equals k the fraction q over r")[$V = k frac(q, r)$], which has a spherical symmetry. Therefore, we use the spherical del operator in the formula #math.equation(block: false, alt: "E → equals − ∇ → V")[$arrow("E") = "−" arrow(∇) V$]. Solution Performing this calculation gives us #math.equation(block: true, alt: "E → equals − open parenthesis r ^ the fraction ∂ over ∂ r plus θ ^ the fraction 1 over r the fraction ∂ over ∂ θ plus φ ^ the fraction 1 over r sin θ the fraction ∂ over ∂ φ close parenthesis k the fraction q over r equals − k q open parenthesis r ^ the fraction ∂ over ∂ r the fraction 1 over r plus θ ^ the fraction 1 over r the fraction ∂ over ∂ θ the fraction 1 over r plus φ ^ the fraction 1 over r sin θ the fraction ∂ over ∂ φ the fraction 1 over r close parenthesis .")[$arrow("E") = "−" ( hat("r") frac(∂, ∂ r) + hat("θ") frac(1, r) #h(0.2em) frac(∂, ∂ θ) + hat("φ") frac(1, r #h(0.2em) "sin" #h(0.2em) θ) #h(0.2em) frac(∂, ∂ φ) ) k frac(q, r) = "−" k q ( hat("r") frac(∂, ∂ r) #h(0.2em) frac(1, r) + hat("θ") frac(1, r) #h(0.2em) frac(∂, ∂ θ) #h(0.2em) frac(1, r) + hat("φ") frac(1, r #h(0.2em) "sin" #h(0.2em) θ) #h(0.2em) frac(∂, ∂ φ) #h(0.2em) frac(1, r) ) .$] This equation simplifies to #math.equation(block: true, alt: "E → equals − k q open parenthesis r ^ the fraction −1 over r squared plus θ ^ 0 plus φ ^ 0 close parenthesis equals k the fraction q over r squared r ^")[$arrow("E") = "−" k q ( hat("r") frac(−1, r^(2)) + hat("θ") 0 + hat("φ") 0 ) = k frac(q, r^(2)) hat("r")$] as expected. Significance We not only obtained the equation for the electric field of a point particle that we’ve seen before, we also have a demonstration that #math.equation(block: false, alt: "E →")[$arrow("E")$] points in the direction of decreasing potential, as shown. #figure(figph[The figure shows a charge Q and radially outward electric field vectors from Q.], alt: "The figure shows a charge Q and radially outward electric field vectors from Q.", caption: [Electric field vectors inside and outside a uniformly charged sphere.]) ] #examplebox("Example 2")[Electric Field of a Ring of Charge][ Use the potential found in to calculate the electric field along the axis of a ring of charge . #figure(figph[The figure shows a ring of charge located on the xy-plane with its center at the origin. Point P is located on the z-axis at distance z away from the origin.], alt: "The figure shows a ring of charge located on the xy-plane with its center at the origin. Point P is located on the z-axis at distance z away from the origin.", caption: [We want to calculate the electric field from the electric potential due to a ring charge.]) Strategy In this case, we are only interested in one dimension, the #emph[z]-axis. Therefore, we use #math.equation(block: false, alt: "E sub z equals minus the fraction ∂ V over ∂ z")[$E_(z) = − frac(∂ V, ∂ z)$] with the potential #math.equation(block: false, alt: "V equals k sub e the fraction q sub tot over the square root of z squared plus R squared")[$V = k_(e) frac(q_("tot"), sqrt(z^(2) + R^(2)))$] found previously. Solution Taking the derivative of the potential yields #math.equation(block: true, alt: "E sub z equals minus the fraction ∂ over ∂ z the fraction k sub e q sub tot over the square root of z squared plus R squared equals k sub e the fraction q sub tot z over open parenthesis z squared plus R squared close parenthesis to the power 3 / 2 .")[$E_(z) = − frac(∂, ∂ z) #h(0.2em) frac(k_(e) q_("tot"), sqrt(z^(2) + R^(2))) = k_(e) frac(q_("tot") z, attach(( z^(2) + R^(2) ), t: 3 "/" 2)) .$] Significance Again, this matches the equation for the electric field found previously. It also demonstrates a system in which using the full del operator is not necessary. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Which coordinate system would you use to calculate the electric field of a dipole? #solutionbox[ Any, but cylindrical is closest to the symmetry of a dipole. ] ] === Summary - Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field. - This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator. === Conceptual Questions If the electric field is zero throughout a region, must the electric potential also be zero in that region? #solutionbox[ No. It will be constant, but not necessarily zero. ] Explain why knowledge of #math.equation(block: false, alt: "E → open parenthesis x , y , z close parenthesis")[$arrow("E") ( x , y , z )$] is not sufficient to determine #emph[V] (#emph[x],#emph[y],#emph[z]). What about the other way around? === Problems Throughout a region, equipotential surfaces are given by #math.equation(block: false, alt: "z equals constant")[$z = "constant"$]. The surfaces are equally spaced with #math.equation(block: false, alt: "V equals 100 V")[$V = 100 #h(0.2em) "V"$] for #math.equation(block: false, alt: "z equals 0.00 m, V equals 200 V")[$z = 0.00 #h(0.2em) "m," #h(0.2em) V = 200 #h(0.2em) "V"$] for #math.equation(block: false, alt: "z equals 0.50 m, V equals 300 V")[$z = 0.50 #h(0.2em) "m," #h(0.2em) V = 300 #h(0.2em) "V"$] for #math.equation(block: false, alt: "z equals 1.00 m.")[$z = 1.00 #h(0.2em) "m."$] What is the electric field in this region? #solutionbox[ The problem is describing a uniform field, so #math.equation(block: false, alt: "E equals 200 V/m")[$E = 200 #h(0.2em) "V/m"$] in the –#emph[z]-direction. ] In a particular region, the electric potential is given by #math.equation(block: false, alt: "V equals − x y squared z plus 4 x y .")[$V = "−" x y^(2) z + 4 x y .$] What is the electric field in this region? Calculate the electric field of an infinite line charge, throughout space. #solutionbox[ Apply #math.equation(block: false, alt: "E → equals − ∇ → V")[$arrow("E") = "−" arrow(∇) V$] with #math.equation(block: false, alt: "∇ → equals r ^ the fraction ∂ over ∂ r plus φ ^ the fraction 1 over r the fraction ∂ over ∂ φ plus z ^ the fraction ∂ over ∂ z")[$arrow(∇) = hat("r") frac(∂, ∂ r) + hat(φ) frac(1, r) #h(0.2em) frac(∂, ∂ φ) + hat("z") frac(∂, ∂ z)$] to the potential calculated earlier, #linebreak() #math.equation(block: false, alt: "V equals −2 k λ ln s : E → equals 2 k λ the fraction 1 over r r ^")[$V = −2 k λ "ln" s : #h(0.2em) arrow("E") = 2 k λ frac(1, r) hat("r")$] as expected. ]