Irodov 3.343: Coils Combine Like Resistors – Amit & Deepali Sinha Foundation

Problem Irodov 3.343

Two identical coils, each with inductance $L$ , are connected together, (a) in series and (b) in parallel. Taking their mutual inductance as negligible, find the inductance of the combination in each case.

(a) The two coils, each of inductance L, in series between two terminals. — (a) The two coils, each of inductance $L$ , in series between two terminals.

(b) The same two coils in parallel between two terminals.

Given

two coils, each $L$
mutual inductance negligible
coil resistance negligible (assumed)

Find

(a)

L_{\mathrm{s}}

in series · (b)

L_{\mathrm{p}}

in parallel

Solution

Step 1What the inductance of a combination means

A combination behaves as a single inductance $L_{\mathrm{tot}}$ if the voltage across its terminals is $L_{\mathrm{tot}}$ times the rate of change of the current entering it. With no mutual inductance, each coil’s voltage is set by its own current alone, and with negligible resistance (assumed) it is

V = L\,\frac{dI}{dt} \tag{1}

Step 2(a) In series

The same current $I$ flows through both coils, and their voltages add. Using (1) for each:

V_{\mathrm{tot}} = L\,\frac{dI}{dt} + L\,\frac{dI}{dt} = 2L\,\frac{dI}{dt}

L_{\mathrm{s}} = 2L

(2)

Step 3(b) In parallel

Now both coils have the same voltage $V$ across them, and the current entering splits as $I = I_1 + I_2$ . By (1) each branch current changes at the rate $V/L$ , so

\frac{dI}{dt} = \frac{dI_1}{dt} + \frac{dI_2}{dt} = \frac{V}{L} + \frac{V}{L} = \frac{2V}{L}

Writing this as $V = L_{\mathrm{p}}\,dI/dt$ :

L_{\mathrm{p}} = \frac{L}{2}

(3)

Check

The energy agrees in both cases. In series each coil carries $I$ , so the stored energy is twice $LI^2/2$ , which is $L_{\mathrm{s}}I^2/2$ with $L_{\mathrm{s}}$ from (2). In parallel identical coils share the current equally, each $I/2$ , so the energy is

2\cdot\frac{L}{2}\left(\frac{I}{2}\right)^2 = \frac{LI^2}{4} = \frac{1}{2}\,\frac{L}{2}\,I^2,

which is $L_{\mathrm{p}}I^2/2$ with $L_{\mathrm{p}}$ from (3).

Remark

Inductances combine like resistances, because each element’s voltage is proportional to a single quantity, the current for a resistor and its rate of change for a coil, and series and parallel connection only add voltages or currents. Coupling changes this: two identical coils in series with mutual inductance $L_{12}$ give $2L \pm 2L_{12}$ , anything from nearly $0$ to $4L$ depending on how they are wound and placed. That is why the statement asks for the coupling to be negligible.