Learning Goal: To better understand self-inductance, using the example of a long

Question

Learning Goal:

To better understand self-inductance, using the example of a long solenoid.

To understand self-inductance, it is helpful to consider the specific example of a long solenoid, as shown in the figure. (Figure 1) This solenoid has radius R and length Z along the z axis, and is wound with n turns per unit length, so that the total number of turns is equal to nZ. Assume that the length of the solenoid is much greater than its radius.

As the current through the solenoid changes, the resulting magnetic flux through the solenoid will also change, and an electromotive force will be generated across the solenoid according to Faraday's law of induction:

E=???total?t,

where ?total is the total magnetic flux passing through the solenoid.

The self-inductance L is defined to be L=?total/I, where I is the current passing through the solenoid. Using the self-inductance, Faraday's law can be rewritten as E=?L?I/?t.

The direction of the emf can be determined using Lenz's law: The induced emf always opposes any change in the current I.

Part A

Within the solenoid, but far from its ends, what is the magnetic field B due to the current I?

Express your answer in terms of some or all of the following variables I, n, Z and any relevant constants (such as ?0).

Part B

What is the magnetic flux ?1 through a single turn of the solenoid?

Express your answer in terms of the magnetic field B, quantities given in the introduction, and any needed constants.

Part C

Suppose that the current varies with time, so that ?I/?t?0. Find the total electromotive force Einduced in the solenoid due to this change in current.

Express your answer in terms of ?I/?t, n, Z, and R.

Part D

The self-inductance L is related to the self-induced emf E by the equation E=?L?I/?t. Find L for a long solenoid. (Hint: The self-inductance L will always be a positive quantity.)

Express the self-inductance in terms of the number of turns per length n, the physical dimensionsR and Z, and relevant constants.

Part E

Which of the following is always a true statement?

Which of the following is always a true statement?

Part F

Suppose a constant current I flows through the inductor, but you are not told whether this current is positive, negative, or zero. Now consider the effect that applying an additional voltage to the inductor will have on the current I already flowing through it; imagine that the voltage is applied to end A, while end B is grounded. Which one of the following statements is true?

Suppose a constant current  flows through the inductor, but you are not told whether this current is positive, negative, or zero. Now consider the effect that applying an additional voltage to the inductor will have on the current  already flowing through it; imagine that the voltage is applied to end A, while end B is grounded. Which one of the following statements is true?

If I is positive, the potential at end A will necessarily be higher than that at end B. If ?I/?t is positive, the potential at end A will necessarily be higher than that at end B. If I is positive, the potential at end A will necessarily be lower than that at end B. If ?I/?t is positive, the potential at end A will necessarily be lower than that at end B.

Explanation / Answer

a) B(t) = mu0 n I(t)

-----------------------------------------------------

b) flux(t) = B(t) A = B pi R2

= mu0 n I(t)  (pi R2)

-----------------------------------------------

c) flux(total) = n Z B pi R2

flux(total) = mu0 n2 I(t) * pi R2 * Z

emf = - d flux(total) / dt

emf = - n2 Z dI(t)/ dt * (mu0 * pi R2)

---------------------------------------------

d) L = n2 Z mu0 * pi R2

--------------------------------------------

e) If dI/dt is positive, the potential at end A will necessarily be higher than that at end B

--------------------------------------------

f) If V is positive, then I could be positive or negative, while dI(t)/dt will necessarily be negative.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.