Distance Formula in Curvilinear Coordinates

• This problem is used in the following sequences

  1. << Electrostatic Potential Due to a Pair of Charges (with Series) | Ring Cycle Sequence | Acting Out Charge Densities >>

• keyboard Electrostatic potential of spherical shell

  keyboard Computational Activity

  120 min.

  Electrostatic potential of spherical shell

  Computational Physics Lab II 2022

  electrostatic potential spherical coordinates

  Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.
• group Electrostatic Potential Due to a Ring of Charge

  group Small Group Activity

  30 min.

  Electrostatic Potential Due to a Ring of Charge

  Static Fields 2023 (8 years)

  electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

  Power Series Sequence (E&M)

  Warm-Up

  Ring Cycle Sequence

  Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.

  In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Box Sliding Down Frictionless Wedge

  group Small Group Activity

  120 min.

  Box Sliding Down Frictionless Wedge

  Theoretical Mechanics (4 years)

  Lagrangian Mechanics Generalized Coordinates Special Cases

  Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
• assignment_ind Curvilinear Coordinates Introduction

  assignment_ind Small White Board Question

  10 min.

  Curvilinear Coordinates Introduction

  Static Fields 2023 (11 years)

  Cylindrical coordinates spherical coordinates curvilinear coordinates

  Curvilinear Coordinate Sequence

  First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles \(\theta\) and \(\phi\). Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• group Electric Field Due to a Ring of Charge

  group Small Group Activity

  30 min.

  Electric Field Due to a Ring of Charge

  Static Fields 2023 (8 years)

  coulomb's law electric field charge ring symmetry integral power series superposition

  Power Series Sequence (E&M)

  Ring Cycle Sequence

  Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

  In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Magnetic Vector Potential Due to a Spinning Charged Ring

  group Small Group Activity

  30 min.

  Magnetic Vector Potential Due to a Spinning Charged Ring

  Static Fields 2023 (6 years)

  compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

  Power Series Sequence (E&M)

  Ring Cycle Sequence

  Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

  In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Magnetic Field Due to a Spinning Ring of Charge

  group Small Group Activity

  30 min.

  Magnetic Field Due to a Spinning Ring of Charge

  Static Fields 2023 (7 years)

  magnetic fields current Biot-Savart law vector field symmetry

  Power Series Sequence (E&M)

  Ring Cycle Sequence

  Students work in small groups to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

  In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• assignment Circle Trig Complex

  assignment Homework

  Circle Trig Complex

  Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2023 (2 years)

  Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

  

• group Static Fields Equation Sheet

  group Small Group Activity

  5 min.

  Static Fields Equation Sheet

  Static Fields 2023 (5 years)
• group Velocity and Acceleration in Polar Coordinates

  group Small Group Activity

  10 min.

  Velocity and Acceleration in Polar Coordinates

  Central Forces 2023 (3 years) Use geometry to find formulas for velocity and acceleration in polar coordinates.
• Static Fields 2023 (6 years)

  The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

  1. Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.

     

  2. Show that this same distance written in cylindrical coordinates is: \begin{equation} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2} \end{equation}
  3. Show that this same distance written in spherical coordinates is: \begin{equation} \left\vert\vec r -\vec r\,{}'\right\vert =\sqrt{r'^2+r\,{}^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}') +\cos\theta\cos\theta\,{}'\right]} \end{equation}
  4. Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

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