accessibility_new Kinesthetic
10 min.
assignment Homework
The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}
assignment Homework
Imagine you're standing on a landscape with a local topography described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Does your result makes sense from the graph?
assignment Homework
Find the gradient of each of the following functions:
assignment Homework
Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\)
and \(y\) are measured in meters and that \(\mu\) is measured in kilograms.
Four points are indicated on the plot.
A contour map for a different function is shown above. On a printout of this contour map, sketch a field vector map of the gradient of this function (sketch vectors for at least 10 different points). The direction and magnitude of your vectors should be qualitatively accurate, but do not calculate the gradient for this function.
assignment Homework
Consider the finite line with a uniform charge density from class.
group Small Group Activity
30 min.
assignment Homework
Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}
group Small Group Activity
30 min.
assignment Homework
Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}
Which pairs (if any) of these vectors
Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates \((x,y)\) measured in miles. Your global positioning system says your present location is at one of the following points (pick one): \[A:(1,4),\quad B:(4,-9),\quad C:(-4,9),\quad D:(1,-4),\quad E:(2,0),\quad F:(0,3)\] Your guidebook tells you that the height \(h\) of the hill in feet above sea level is given by \[h=a-bx^2-cy^2\] where \(a=5000~\mathrm{ft}\), \(b=30\,\mathrm{\frac{ft}{mi^2}}\), and \(c=10\,\mathrm{\frac{ft}{mi^2}}\).