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Gradient

FIXME

In-class Content

  • QUIZ (10 min)
  • The Hillside (SGA - 20 min)
  • Acting Out the Gradient (Kinesthetic Activity - 10 min)
  • Visualizing Gradient (Maple/Mathematica - 15 min)
  • (Lecture - 15 min)
    • The gradient in different coordinate systems.
    • E is the gradient of the potential.
    • Finding the gradient of the potential due to a point charge at an arbitrary location using the dot product.
      • Remind students that the dot product can be foiled.

Optional In-class content

Homework for Symmetries

  1. (DirectionalDerivative)

    Imagine you're standing on a landscape with a local topology described by the function $f(x, y)= k x^{2}y$, where $k=20 \frac{m}{km^3}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot (3 km,2 km) and there is a cottage located at (1 km, 2 km). At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function $f(x, y)$ in Mathematica. Does your result makes sense from the graph?

  2. (GradientPractice)

    Find the gradient of each of the following functions:

    1. $$f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z}$$

    2. $$\sigma(\theta,\phi)=\cos\theta \sin^2\phi$$

    3. $$\rho(s,\phi,z)=(s+3z)^2\cos\phi$$

  3. (GradientPtChargeOrigin)

    The electrostatic potential due to a point charge at the origin is given by: $$V=\frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

    1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.

    2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.

    3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

Homework for Symmetries

  1. (GradientPtCharge)

    Consider the fields at a point $\rr$ due to a point charge located at $\rr'$.

    1. Write down an expression for the electrostatic potential $V(\rr)$ at a point $\rr$ due to a point charge located at $\rr'$. (There is nothing to calculate here.)

    2. Write down an expression for the electric field $\EE(\rr)$ at a point $\rr$ due to a point charge located at $\rr'$. (There is nothing to calculate here.)

    3. Working in rectangular coordinates, compute the gradient of $V$.

    4. Write several sentences comparing your answers to the last two questions.

  2. (LineSources)

    Consider the fields around both finite and infinite uniformly charged, straight wires.

    1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the expression for the electrostatic potential that we found in class:

      $$V(\Vec r)={2\lambda\over 4\pi\epsilon_0}\, \ln{ r_0\over r}$$

      Compare your result to the solution found from Coulomb's law. Which method is easier?

    2. Find the electric field around a finite, uniformly charged, straight wire, at a point a distance $r$ straight out from the midpoint, starting from the expression for the electrostatic potential that we found in class:

      $$V(\Vec r)={\lambda\over 4\pi\epsilon_0} \left[\ln{\left(L + \sqrt{L^2+r^2}\right)}- \ln{\left(-L + \sqrt{L^2+r^2}\right)}\right]$$

      Compare your result to the solution found from Coulomb's law. Which method is easier?

    3. Find the electric field around an infinite, uniformly charged, straight wire, starting from Coulomb's Law.

    4. Find the electric field around a finite, uniformly charged, straight wire, at a point a distance $s$ straight out from the midpoint, starting from Coulomb's Law.

  3. (ConductorsGem235ModB)

    Find $E_r$, the radial component of the electric field and plot it as a function of $r$. %Are the discontinuities in the electric field what you expect from our unit on boundary conditions? Explain.


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