This is a closed book exam; the formula sheet provided is the only supplemental material permitted on this exam.

Programmable calculators and graphing calculators may be used during this exam, but not for storage of additional notes or formulas.

Since this exam booklet may be separated for grading; it is important to:

Show ALL work on problem sheet.

Please read questions carefully.

Credit may be lost inadvertently if solutions are not neat and orderly.

Be careful with units, signs, and significant figures.

1. (20 points)

1. A charged particle is circulating clockwise in an external magnetic field pointing away from you. Is it positively or negatively charged? Briefly explain and include sketch.

   
   

2. Two neighboring turns of wire are carrying identical currents (such as consecutive turns of wire in a solenoid). Is the force they experience repulsive or attractive? Explain.

    

3. A capacitor charged by an emf in series with a resistor stores only half the energy that the emf delivers. What happens to the rest of the energy? In words...

   
   

4. A wire loop is near a long wire carrying current in the direction shown. If the current in the long wire decreases, which direction will the current be induced in the loop?

    

5. State Lenz’s law in words.

2. (40 points)

Consider a toroid of square cross section, with inner radius a = 2.0 cm and outer radius b = 3.0 cm, consisting of 125 turns of 18-gauge wire. In a recent quiz, you determined that the magnetic field inside an energized toroid is where r is the distance from the axis. From that result we derived that the inductance of a rectangular toroid is
Evaluate the inductance and resistance of this toroid. (The resistance per unit length of the wire is 205 micro-ohm/cm.) What magnetic field will be created in the center of the toroid (point shown) after it has been energized by a 1.0 V emf? What is the maximum rate of change of current as the toroid is being energized?
If a separate turn of 18-gauge wire of radius 1.0 cm had been added around the toroid as shown, what maximum current would be induced in it as the toroid is energized? What is the magnitude of the magnetic field created at the center of the circular turn because of this induced current?

3. (20 points)

Given a long straight wire of radius a carrying a uniformly-distributed current, apply Ampere’s law to find an expression for the magnetic field inside\ the wire at a distance r from its center. Please include a sketch showing the wire and direction of current, magnetic field lines with directions, and the amperian loop used.

4. (20 points)

Recently in class the above result was used to determine the contribution of the "internal" field to the self-inductance. By using energy considerations I showed that its contribution per unit length was

Rather than repeat that same calculation, consider instead the case of a wire with a non-uniform current density given by j = j₀ r/a. In one of our homework exercises, it was shown that the magnetic field inside the wire could be expressed as

Use this result (no need to rederive it!) to evaluate the "internal" contribution to the inductance per unit length in this "unusual" wire. How does it compare to the value for an ordinary wire?