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Problem 28.9: Slinky Solenoid

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A solenoid is made by wrapping a long wire many times around a cylinder. This animation shows a cross section of the solenoid (cylinder). Each loop of the wire circles behind and in front of the computer screen, so your view of the solenoid is a long tube sliced in half, lying on its side (position is given in centimeters and magnetic field is given in millitesla). Use the slider to change the current through the wire. This solenoid has a fixed number of wire coils, but by using the slider, you can stretch it or compress it (think of a solenoid made from a Slinky®, in which the ends of the Slinky® are connected to a current source). Restart.

  1. The black box is an Amperian loop. For which sides of the box is ∫ B · dl = 0? Why? For the other side, show that ∫ B · dl = BL, where L is the length of the side and B is the magnitude of the magnetic field at that point. How much current is enclosed in the Amperian loop?
  2. Therefore, how many loops/centimeter are there?
  3. How many total loops are there in this solenoid (how many coils in the Slinky®)?
  4. Develop an expression for the magnetic field as a function of the length of the solenoid (same number of loops as the solenoid is stretched or compressed).

Problem authored by Anne J. Cox.

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