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Section 11.5 Assignment 5

  1. Assume that there is a particle with charge \(q\) located at everyapoint in \(RR^3\) which has integer coordinates. What is the flux of the electric field over the following surfaces? (8)

    1. The sphere of radius \(\frac{1}{2}\) centred at the origin.

    2. The sphere of radius \(\frac{1}{4}\) centred at the point \(\left(0, 0, \frac{1}{2}\right)\text{.}\)

    3. The sphere of radius \(1\) centred at the the point \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)\text{.}\)

    4. The sphere of radius \(\frac{5}{2}\) centred at the origin.

  2. Let \(\sigma\) be the horizontal solid disc of radius \(1\) centre about the \(z\) axis in the plane \(z=1\text{.}\) Consider a current density in the neighbourhood of this disc. (16)

    \begin{equation*} J = \frac{\sin t}{\mu_0} (z, 0 0) \end{equation*}

    1. Calculate \(I_\sigma\text{.}\)

    2. Calculate the induced magnetic field. (You have some choice here: you need any field with a particular curl. Chose the most straightforward vector potential and then check that \(\nabla \cdot B = 0\) is still satisfied. As a hint, you can find a field where both \(B_1\) and \(B_3\) are zero.)

    3. Calculate \(\Phi_{B,\sigma}\text{.}\)

    4. Calculate the line integral of \(B\) around the circle \(\del \sigma\) in two says: directly and using Maxwell's equations. (Make sure, of course, that the two calculations agree.)

  3. On an open set \(U\) in the first octant of \(\RR^3\text{,}\) consider the following forms and vector fields. (Implicitly, this open set is treated as a manifold with coordinate function \(\phi = \Id\text{.}\) (16)

    \begin{align*} v_1 \amp = (x^2 + y^2) \del_x + (x^2 + z^2) \del_y + (y^2 + z^2) \del_z \\ v_2 \amp = x \del_x + y \del_y + z \del_z \\ v_3 \amp = \frac{1}{xyz} \del_x + \frac{1}{xyz} \del_y + \frac{1}{xyz} \del_z \\ w_1 \amp = x dx + y dx + z dz \\ w_2 \amp = xy dx \wedge dy + yz dx \wedge dz + z^2 dy \wedge dz \\ w_3 \amp = \frac{1}{x^2 + y^2 + z^2} dx \wedge dy \wedge dz \end{align*}

    Do the following calculations.

    1. \begin{equation*} w_1(v_1) \end{equation*}

    2. \begin{equation*} w_1(v_2) \end{equation*}

    3. \begin{equation*} i_{v_1} w_2 \end{equation*}

    4. \begin{equation*} w_1 \wedge w_2 \end{equation*}

    5. \begin{equation*} i_{v_3} w_2 \end{equation*}

    6. \begin{equation*} dw_2 \end{equation*}

    7. \begin{equation*} dw_1 \end{equation*}

    8. \begin{equation*} w_3 \wedge w_1 \end{equation*}

  4. In several parts, this question looks at the generlization of curl to \(\RR^4\text{.}\) (8)

    1. Let \(\omega\) be a 1-form on an open set in \(\RR^4\text{.}\) Calculate \(d\omega\text{.}\) Compare the components to the components of curl.

    2. Let \(\omega\) be a 2-form on an open set in \(\RR^4\text{.}\) Calculate \(d\omega\text{.}\) Compare the components to the components of curl.

    3. Why are both of these calculations a generalization of curl?