CP CALC In the circuit in Fig. P29.49, an emf of 90.0 V is added in series with the capacitor an

step 1 So for this problem, we have the same setup as an earlier problem. So make sure to refer to that

CP CALC In the circuit in Fig. P29.49, an emf of 90.0 V is...

CP CALC In the circuit in Fig. P29.49, an emf of 90.0 $\mathrm{V}$ is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor and the switch, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in Problem $29.49 .$ The switch is closed at $t=0 .$ When the current in the large circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit?

CP CALC In the circuit in Fig. P29.49, an emf of 90.0 $\mathrm{V}$ is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor
and the switch, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in Problem $29.49 .$ The switch is closed at $t=0 .$ When the current in the large
circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit?

Key Concepts

Transient Response in RC Circuits

The transient behavior of RC circuits, including the charging and discharging of a capacitor when a switch is closed, is crucial in such problems. It helps in understanding how current changes over time immediately after closing the switch, which is necessary for determining the rate of change of current that drives the induced emf in the nearby circuit.

Faraday's Law of Electromagnetic Induction

This law states that a changing magnetic flux through a circuit induces an electromotive force (emf) in that circuit. It is fundamental in understanding how a time-varying current in one circuit can create an induced emf in a nearby circuit, leading to an induced current.

Lenz's Law

Lenz's Law provides the direction of the induced current, stating that the induced emf always acts to oppose the change in magnetic flux that produced it. This concept is used to determine whether the induced current will circulate clockwise or counterclockwise in order to oppose the increase or decrease in the magnetic field.

Mutual Inductance

Mutual inductance is the phenomenon by which a changing current in one circuit induces an emf in a neighboring circuit. It quantifies the degree of coupling between two circuits and directly relates the rate of change of current in the primary circuit to the induced emf in the secondary circuit.

Transcript

00:01 So for this problem, we have the same setup as an earlier problem.

00:06 So make sure to refer to that problem because we're going to use that result for this one.

00:12 The only difference for this new one is that now we have a battery.

00:18 So there's a battery here of 90 volts.

00:21 And when the switch closes, we're going to have not a discharging capacitor, but a charging capacitor.

00:27 However, the result for the current is going to be identical.

00:36 It's actually the exact same formula.

00:37 So this is what we found for the current in the small loop.

00:46 What we found was n -mu -0 -b over 2 -p times the lawn of 1 plus a over c times i over rc times i over rc, times 1 over r small, where now i is 5 .00 amps, and r small was 15 oms.

01:19 So it's the exact same result because the exact same physics is involved.

01:27 D -i -d -t is still going to be i over rc, so it doesn't matter that it's charging versus discharging, because you'll see when you take the derivative that the exact same functional form is found.

01:43 So when you take the derivative of 1 minus e to the negative t over r not c, that's still 1 over r0c times e to the negative t over r not c.

02:00 So it's the exact same formula.

02:05 So once you substitute in the given numbers, you should find that this is 1 .83 times 10 to the minus 3 amperes.

02:18 Again, so refer to the problem that's referenced in this question to see where this formula comes from, but the point is it's the exact same formula...

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