Question

Learning Goal:

To leam about self-inductance from the example of a long solenoid.

To explain self-inductance, it is helpful to consider the specific example of a long solervid, as shown in the figure. This solenoid has only one winding, and so the EMF induced by its changing current appoars across the solenoid itsel. This contrasts with mutual inductance, where this voltege appears across a second coil wound on the same cylincter as the lirst.

Part A

Suppose that the current in the solenoid is I(t). Within the solenoid, but far from its ends, what is the magnetic field B(t) due to this current?

Part B 

What is the magnetic flux φ₁(t) through a single tum of the solenoid? 

Part C 

Suppose that the current varies with time, so that dI(t)/dt≠0. Find the electromotive force ε induced across the entire solenoid due to the change in current through the entire solenoid. 

Part D 

The self-inductance L is related to the self-induced EMR ε(t) by the equation ε(t) = -Ldl(t) /dt. Find L for a long solenoid. 

Now consider an inductor as a circuit element. Since we are now treating the inductor as a circuit element, we must discuss the voltage across it, not the EMF inside it. The important point is that the inductor is assumed to have no resistance. This means that the net electric field inside it must be zero when it is connected in a circuit. Otherwise, the current in it will become infinite. This means that the induced electric field \(\vec{E}_{\mathrm{n}}\) deposits charges on and around the inductor in such a way as to produce a nearly equal and opposite electric field \(\vec{E}_{\mathrm{c}}\) such that \(\vec{E}_{\mathrm{c}}+\overrightarrow{E_{\mathrm{n}}} \rightarrow 0 !\) Kirchhoff's loop law defines voltages only in terms of fields produced by charges (like \(\vec{E}_{\mathrm{c}}\) ), not those produced by changing magnetic fields (like \(\vec{E}_{\mathrm{n}}\) ). So if we wish to continue to use Kirchhoff's loop law, we must continue to use this definition consistently. That is, we must define the voltage \(V_{\mathrm{AB}}=V_{\mathrm{A}}-V_{\mathrm{B}}=+\int_{\mathrm{A}}^{\mathrm{B}} \vec{E}_{\mathrm{c}} \cdot d \vec{l}\) alone (note that the integral is from \(\mathrm{A}\) to \(\mathrm{B}\) rather than from \(\mathrm{B}\) to \(\mathrm{A}\), hence the positive sign). So finally, \(V_{\mathrm{AB}}=\int_{\mathrm{A}}^{\mathrm{B}} \vec{E}_{c} \cdot d \vec{l}=\int_{\mathrm{A}}^{\mathrm{B}}-\vec{E}_{n} \cdot d \vec{l}=-\mathcal{E}=+L \frac{d I(t)}{d t}\), where we have used \(\vec{E}_{c}+\vec{E}_{n}=0\) and the definition of \(\mathcal{E}\).

Part E

Which of the following statements is true about the inductor in the figure in the problem introduction, where I(t) is the current through the wire? 

• If I(t) is positive, the voltage at end A will necessarily be greater than that at end B 

• lf dI(t)/dt is positive, the voltage at end A will necessarily be greater than that at end B 

• If I(t) is positive, the voltage at end A will necessarily be less than that at end B. 

• If dI(t)/dt is positive, the voltage at end A will necessarily be less than that at end B. 

Part F 

Now consider the effect that applying an additional voltage to the inductor will have on the current I(t) 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 V is positive, then I(t) will necessarily be positive and dI(t)/dt will be negative 

•  lf V is positive, then I(t) will necessarily be negative and di(t)/dt will be negative. 

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

•  If V is positive, then I(t) will necessarily be positive and dI(t)/dt will be positive 

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

•  If V is positive, then It) will necessarily be negative and dI(t)/dt will be positive

Answer 1