Question about current decay in R-L circuit

[issacnewton]

Hello

I am reading Young, Freedman's University physics and I have some questions about the
sign conventions followed in getting the equations for the current decay in the R-L circuit.
I have attached two snapshots which will help the discussions. Consider the series R-L circuit
as shown in the diagram. Say switch S₁ is closed and switch S₂ is open for a long time. So that means steady current has been established in the upper part of
the circuit. Now open S₁ and close S₂. Now authors give the following
equation for the first case, when the current is building.

$$\mathcal{E}-iR -L\frac{di}{dt}=0$$

which makes sense using the second diagram I attached. But the authors say that , for the
second case , when the current is decaying through the bottom part of the circuit, the equation to be solved becomes,

$$-iR -L\frac{di}{dt}=0$$

which doesn't make sense. Since the current is now decreasing, going around the loop
in the direction of current (assuming that the direction of the current is still the same),
and using the second diagram which I have uploaded, the Kirchhoff's rule says that

$$-iR +L\frac{di}{dt}=0$$

But this doesn't give correct final result for the decay. So am I doing something wrong ?

thanks

 

[Gordianus]

The minus sign in L di/dt is a consequence of Lenz's law (or the minus sign in Faraday's). Just because we know the current will decrease, we don't change the sign because that would give us an ever increasing current (contrary to our expectations)

 

[issacnewton]

But Gordianus, but is there not a potential gain in such situation of we trace the loop in the same direction as the decaying current. I was trying to follow the rules of Kirchhoff using the second diagram given for the situation where $\frac{di}{dt} <0$.

 

[Gordianus]

Exactly. When di/dt<0 you want a positive voltage drop. If you multiply by -L you obtain a positive number. On the contrary, if you multiply by L, you obtain a negative number (this isn't what you want).
In a nut, you don't have to switch the sign from positive to negative. The same equation gives the right answer for both cases.

 

[issacnewton]

Gordianus, I think it makes sense now... So are the signs used the way they are used to get the end equations working ?

 

[Gordianus]

The safe method is this:
a) You assume the current flows in some direction (it doesn't matter which one).
b) You apply KVL moving in the same direction current does.

With these previous assumptions the voltage drop across the inductance is -L di/dt, whether the current grows or decays.

 

[issacnewton]

ok makes sense