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Circuit and input waveforms

So in this circuit, initially the current and voltage is 0. At t = 0 the voltage snaps to V.

Now my analysis is: According to Lenz Law the voltage developed will be in the direction as specified in red markings in the image because a back emf will be developed. So the KVL equation would be:

\$-v_S + i_L R - v_L = 0\$

Applying Faraday's Law we get:

\$-v_S + i_L R - L \frac{di_L}{dt} = 0\$

Now I have taken \$v_L\$ to be the magnitude of the voltage, because since we indicated the direction in the circuit.

But the textbook has this equation, obviously I am wrong but I cannot see where my reasoning is wrong.

Equation

Allen
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1 Answers1

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Lenz's Law says that the direction of an induced current is always such as to oppose the change in the magnetic field that produces it.

In your circuit the current producing the magnetic field is entering the coil at the '+' end, so the induced current must be flowing in the opposite direction, out of the '+' end. For current to be pushed out of that end the induced voltage must be positive, not negative.

Bruce Abbott
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  • But doesn't current flow from positive to negative, and if the markings in the text is correct won't it cause the induced current to flow in the same direction of the current which changes the field (which is \$i_L\$) – Allen Dec 28 '19 at 18:46
  • @Allen, that rule doesn't apply to an inductor. For an inductor, the direction of current is independent of the direction of the appled voltage. It's the change (derivative) of the current that depends on the voltage. – The Photon Dec 28 '19 at 19:08
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    Lenz's Law says the induced current flows in the **opposite** direction to the current causing it. This is how it opposes the current causing the magnetic field to increase. If it was the other way around (induced current went in the same direction) it would strengthen the magnetic field, causing even more induced current and so on to infinity. – Bruce Abbott Dec 28 '19 at 19:12
  • @BruceAbbott But if the positive and negative parts of the inductor are as it is in the text (which are in black) won't the back emf cause the current to flow in the same direction as \$i_L\$ inside the inductor. – Allen Dec 28 '19 at 19:35
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    The increasing current flowing into the coil causes an induced voltage V=L*di/dt which pushes back on it just enough to keep the current rise down to the amount required to satisfy the formula. Imagine if it was the other way around - negative induced voltage pulling more current into the coil, creating a faster changing magnetic field which increases the current even more rapidly without limit. The circuit would blow up almost instantly! – Bruce Abbott Dec 28 '19 at 20:36