Consider an electrical circuit containing a two way switch S. Initially S is open and then T1 is con

Q: Why does the inductor act as a short circuit in the steady state?

In a DC circuit, the current eventually becomes constant. Since the voltage across an inductor is proportional to the rate of change of current ($V = L \frac{di}{dt}$), a constant current means zero voltage drop, making it act like a plain wire.

Q: Why doesn't the current drop to zero immediately when the battery is disconnected?

An inductor stores energy in its magnetic field. When the circuit changes, this magnetic field collapses, inducing an electromotive force (EMF) that keeps the current flowing in the same direction for a short time.

Q: What happens to the energy stored in the inductor?

The energy stored in the inductor's magnetic field ($\frac{1}{2}LI^2$) is gradually dissipated as heat in the $3\,\Omega$ resistor as the current exponentially decays to zero.

Q: Would the answer change if the inductor had some internal resistance?

Yes! If the inductor had internal resistance, the initial steady-state current would be lower, as the total resistance in the left loop would be $R + R_{\text{internal}}$.

Q: How does the current behave after the switch is moved to $T_3$?

The current will exponentially decay to zero according to the formula $I(t) = I_0 e^{-\frac{r}{L}t}$, where $I_0$ is the initial current of $1\text{ A}$.

Question

Consider an electrical circuit containing a two way switch $S$. Initially $S$ is open and then $T_1$ is connected to $T_2$. As the current in $R = 6 \Omega$ attains a maximum value of steady state level, $T_1$ is disconnected from $T_2$ and immediately connected to $T_3$. Potential drop across $r = 3 \Omega$ resistor immediately after $T_1$ is connected to $T_3$ is ……… V. (Round off to the nearest integer)

Sigma Percentile · Accepted Answer

### The Setup: A Tale of Two Loops

Imagine a circuit divided into two distinct neighborhoods, with an inductor acting as the bridge between them. On the left, we have a powerful $6	ext{ V}$ battery and a $6\,\Omega$ resistor. On the right, a solitary $3\,\Omega$ resistor waits patiently. The switch $S$ is the gatekeeper, deciding which neighborhood the inductor gets to interact with.

Initially, the gatekeeper connects the inductor to the left neighborhood by linking $T_1$ to $T_2$. The circuit is now a simple series loop containing the battery, the $6\,\Omega$ resistor, and the inductor.

### The Steady State: Patience is a Virtue

When a DC voltage is first applied to an inductor, it fights back, resisting the flow of current. But as time passes, it slowly gives up the fight. After a long time, the circuit reaches a **steady state**. In this state, the inductor acts just like a plain, zero-resistance wire.

Because the inductor offers no resistance in the steady state, the current flowing through the left loop is determined entirely by the battery and the resistor. Using Ohm's law, we can easily calculate this maximum steady current:

$$I = \frac{V}{R} = \frac{6}{6} = 1	ext{ A}$$

So, a steady current of $1	ext{ A}$ is happily flowing through the inductor, storing energy in its magnetic field.

### The Switch: A Sudden Change

Suddenly, the gatekeeper flips the switch! $T_1$ is disconnected from $T_2$ and immediately connected to $T_3$. The inductor is abruptly cut off from the battery and the $6\,\Omega$ resistor. It is now trapped in a new, closed loop with only the $3\,\Omega$ resistor for company.

### The Inductor's Memory: Defying Change

Here is where the magic of physics happens. An inductor has a kind of "electrical inertia"—it absolutely hates sudden changes in current. The moment the switch is flipped, the magnetic field around the inductor begins to collapse, acting like a temporary battery to keep the current flowing exactly as it was.

Therefore, the current immediately after the switch is thrown ($t = 0^+$) must be identical to the current just before the switch was thrown ($t = 0^-$).

$$I_{	ext{initial}} = 1	ext{ A}$$

### The Final Calculation: Reaping the Rewards

This $1	ext{ A}$ current is now forced to flow through the right loop, passing directly through the $r = 3\,\Omega$ resistor. The question asks for the potential drop across this resistor at this exact moment.

Armed with the current and the resistance, we call upon Ohm's law one last time:

$$V_r = I_{	ext{initial}} 	imes r$$

$$V_r = 1 	imes 3 = 3	ext{ V}$$

The potential drop across the $3\,\Omega$ resistor immediately after the switch is connected to $T_3$ is exactly **$3	ext{ V}$**.

Enter Numerical Value:

Visualized Solution

$Visualizing the Circuit$

$The Steady State$

$Calculating Initial Current$

$Switching the Circuit$

$The Inductor’s Memory$

$Potential Drop Setup$

$Final Calculation$

$The Way Forward$

The Sigma Insight: Self and Mutual Inductance

The Setup

The Steady State

The Switch

The Inductor's Memory

The Final Calculation

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Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Steady State

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Initial Current

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Switching the Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Inductor’s Memory

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Potential Drop Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Calculation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward