Homework for Static Fields

  1. (TrigParameters–Practice) This is a simple practice problem that can be used as preparation for the next problem–Thetaparameters.

    Make sketches of the following functions, by hand, all on the same axes. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& \sin x\\ y &=& 2+\sin x\\ y &=& \sin(2+x)\\ y &=& 2\sin x\\ y&=& \sin 2x \end{eqnarray}

  2. (ThetaParameters) Graph various versions of the theta function to explore how different parameters of the function affect the shape of the graph.

    The function $\theta(x)$ (the Heaviside or unit step function) is a defined as: \begin{eqnarray*} \theta(x) = \left\{ \begin{array}{l l} 1 & \quad \mbox{for $x>0$}\\ 0 & \quad \mbox{for $x<0$}\\ \end{array} \right. \end{eqnarray*} This function is discontinuous at $x=0$ and is generally taken to have a value of $\theta(0)=1/2$.

    Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& \theta (x)\\ y &=& 2+\theta (x)\\ y &=& \theta(2+x)\\ y &=& 2\theta (x)\\ y &=& \theta (2x) \end{eqnarray}

  3. (TriangleParameters) Graph various versions of the triangle function (written with theta functions) to explore how different parameters of the function affect the shape of the graph.

    Consider the function: \begin{eqnarray*} f(x) = 3x\,\theta(x)\,\theta(1-x)+(6-3x)\,\theta(x-1)\,\theta(2-x) \end{eqnarray*} Make sketches of the following functions, by hand, on the axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{eqnarray} y &=& f(x)\\ y &=& 2+f(x)\\ y &=& f(2+x)\\ y &=& 2f(x)\\ y &=& f(2x) \end{eqnarray}

  4. (Delta) Examine a linear charge density written so as to contain delta functions and calculate the total charge.

    The linear charge density from a series of charges along the $x$-axis is given by: $$\lambda(x) = \sum_{n=0}^{10} q_0 \, n^2 \delta\!\left(x-{n\over 10}\right)$$

    1. Write a sentence or two indicating describing the dimensions of each term in this equation, including any constants (for which the dimensions have not been indicated).

    2. What is the total charge on the $x$-axis?

  5. (SlabMass) Find the total mass of several different slabs where the mass density is written in with theta or delta functions. Compare the different solutions to get a physical feel for the meaning of theta and delta functions.

    Determine the total mass of each of the slabs below.

    1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $\rho=A\pi\sin(\pi z/h)$.

    2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}

    3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.

    4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.

    5. What are the dimensions of $A$?

    6. Write several sentences comparing your answers to the different cases above.


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