The ratio b a for a toroidal inductor.

Question

At t = 0, the open switch in Figure P31.46 is thrown closed. We wish to find a symbolic expression for the current in the inductor for time t> 0. Let this current be called i and choose it to be downward in the inductor in Figure P31.46. Identify i, as the current to the right through R, and iz as the current downward through R. (a) Use Kirchhoff's junction rule to find a relation among the three currents. (b) Use Kirchhoff's loop rule around the left loop to find another relationship. (c) Use Kirchhoff's loop rule around the outer loop to find a third relationship. (d) Eliminate i, and i, among the three equations to find an equation involving only the current i. (e) Compare the equation in part (d) with Equation 31.6 in the text. Use this comparison R Figure P31.46 to rewrite Equation 31.7 in the text for the situation in this problem and show that i(t) R, where R' = R,R,/(R, + R,).

Answer 1

Question

Chapter 31, Problem 41AP

To determine

The ratio ba for a toroidal inductor.

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At t = 0, the open switch in Figure P31.46 is thrown closed. We wish to find a symbolic expression for the current in the inductor for time t> 0. Let this current be called i and choose it to be downward in the inductor in Figure P31.46. Identify i, as the current to the right through R, and iz as the current downward through R. (a) Use Kirchhoff's junction rule to find a relation among the three currents. (b) Use Kirchhoff's loop rule around the left loop to find another relationship. (c) Use Kirchhoff's loop rule around the outer loop to find a third relationship. (d) Eliminate i, and i, among the three equations to find an equation involving only the current i. (e) Compare the equation in part (d) with Equation 31.6 in the text. Use this comparison R Figure P31.46 to rewrite Equation 31.7 in the text for the situation in this problem and show that i(t) R, where R' = R,R,/(R, + R,).

You are working on an LC circuit for an experiment you are performing in your basement. You have an appropriate capacitor, but you need to build your own inductor. You wish to cut a wooden ring with a rectangular cross section, as shown, from wood with thickness h = 1.00 cm. You want to wrap 500 turns of wire around it to form a toroidal inductor. For your experiment, you need to have 1.82 × 10–4 J of energy stored in the inductor when it carries a current of 2.00 A. In order to cut the appropriate wooden ring, you need to determine the ratio b/a. Ignore any effect of the wood core on the magnetic field.

You are working on an LC circuit for an experiment you are performing in your basement. You have an appropriate capacitor, but you need to build your own inductor. You wish to cut a wooden ring with a rectangular cross section, as shown, from wood with thickness h. You want to wrap N turns of wire around it to form a toroidal inductor. For your experiment, you need to have energy UB stored in the inductor when it carries a current i. In order to cut the appropriate wooden ring, you need to determine the ratio b/a. Ignore any effect of the wood core on the magnetic field.