Physics 308 Homework
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Homework #10: Due Saturday, April 12, by 12:00noon.

  1. Introduction: Earth's magnetic field has a main dipole part plus a number of lesser-magnitude parts (quadrupole, octopole, etc.). These are visualized here: http://en.wikipedia.org/wiki/Earth's_magnetic_field. Note the image at the upper right looks a bit like the usual dipole, but there are a lot of extra twists/turns in the field lines. This main dipole field may be treated as arising from a single current-carrying loop with perhaps over a billion amps of current. This loop comprises Earth's liquid outer core with an average radius of around 2,400km. You are going to model this.
    Assignment:
    1. Using the Biot-Savart relation, derive the infinitesimal magnetic field element dB at an arbitrary field point (xf, yf, zf). You must use the coordinates (x, y, z) where the single current loop of radius R=2,400km lies in the x-y plane, and the z-axis points to Earth's north geographic pole. The current will run from east to west in this corrdinate system.
    2. Do the integrals that arise from this dB in order to get the expression for the magnetic field at an arbitrary field point (xf, yf, zf).
    3. Plot this net field B assuming the total current is 109A. Plot this out to a distance of 25,000km on each axis.
  2. An aluminum strip 2.0cm wide and 2.0mm thick is placed in a magnetic field with B=3.50Tesla. A current of 4.25A is set up in the strip. The strip is oriented along the x-axis with the current running towards the negative x-axis. The magnetic field points in the negative z direction.
    1. Draw this situation, labeling the direction of the current and the sides of the aluminum strip that becomes positively and negatively charged.
    2. What is the magnitude of the Hall field E and Hall voltage V (or "Hall potential"=Φ) that appears across the strip?
  3. A wire has a current of 2.40A running through it in the z-direction. A square loop with sides 15cm long is located 10cm away from the wire (i.e. the near side of the loop is 10cm away, and the far side is 25cm away). The loop lies in the x-z plane. Calculate the net magnetic flux through the loop. Hint: Note that the magnetic field through the loop is not uniform.
  4. The nearby Claytor Lake hydroelectric power plant has large coils of wire spinning in magnetic fields created by siphoning off some of the electricity generated by the coils themselves. Asume those magnetic fields have a magnitude of 0.025T. The coils of wire have a radius of 1.60m. They produce electricity at a maximum of 220kV. Find the number of current-carrying loops in each set of coils. Recall that you will have to make a reasonable assumption.
  5. A set of wire coils like we have in the intro physics lab has N=120 loops, a radius of R=20cm, and would produce a magnetic field at its center of B=Nμ0I/2R. (a) Derive the epxression for the self-inductance of this set of coils. (b) Find the self-inductance in units of Henrys.
   That's all for homework #10.