An ideal coil of 10 H is connected in series with a resistance of 5 ohm and a battery of 5 V. After

Q: Why doesn't the current reach its maximum value instantly when the switch is closed?

An inductor opposes any change in current. When you close the switch, the sudden attempt to increase the current creates a changing magnetic field, which induces a 'back EMF' that fights the battery's voltage. This forces the current to rise gradually.

Q: What exactly is the time constant ($\tau$)?

The time constant $\tau = L/R$ is a measure of how 'sluggish' the circuit is. Physically, it is the time required for the current to reach approximately $63.2\%$ of its final maximum value.

Q: What happens to the inductor after a very long time ($t \to \infty$)?

After a long time, the current becomes steady and stops changing. Since the induced EMF depends on the *change* in current ($\frac{dI}{dt}$), the back EMF becomes zero. The ideal inductor then acts just like a plain, zero-resistance wire.

Q: Why is the maximum current $I_0$ calculated just using Ohm's law ($E/R$)?

Because in the steady state, the inductor offers no opposition to the constant direct current. The only thing limiting the current is the ohmic resistance $R$ in the circuit.

Q: What does $e^{-1}$ mean in a practical sense?

The mathematical constant $e$ is approximately $2.718$. Therefore, $e^{-1} = \frac{1}{e} \approx 0.368$. So, $1 - e^{-1} \approx 0.632$. This means the current is at $63.2\%$ of its maximum capacity.

Question

An ideal coil of $10	ext{ H}$ is connected in series with a resistance of $5\,\Omega$ and a battery of $5	ext{ V}$. After $2	ext{ s}$, the connection is made, the current flowing (in ampere) in the circuit is

Options:

(A) $1 - e$

(B) $e$

(C) $e^{-1}$

(D) $1 - e^{-1}$

Sigma Percentile · Accepted Answer

### The Stubborn Nature of Inductors

Imagine you are trying to push a heavy boulder. When you first apply force, it doesn't instantly jump to a high speed; it takes time to accelerate due to its inertia. In the world of electronics, an inductor acts very much like that boulder. It possesses 'electrical inertia.'

When you close a switch to connect a battery to a circuit containing an inductor, the current wants to flow immediately. However, the inductor—a coil of wire—detects this sudden change in current. According to Faraday's Law of Induction and Lenz's Law, the changing magnetic field inside the coil induces a 'back EMF' (electromotive force) that actively opposes the rise in current. This is why the current in an $L-R$ (Inductor-Resistor) circuit doesn't spike instantly but instead grows smoothly over time.

### Setting Up the Differential Equation

To understand this mathematically, we turn to Kirchhoff's Voltage Law (KVL). If we traverse the closed loop of our circuit, the sum of the potential differences must be zero. The battery provides a voltage $E$, the resistor drops the voltage by $IR$ (Ohm's Law), and the inductor drops the voltage by $L \frac{dI}{dt}$.

This gives us the differential equation:
$$E - IR - L \frac{dI}{dt} = 0$$

Solving this first-order linear differential equation with the initial condition that $I(0) = 0$ yields the famous exponential growth equation:
$$I(t) = I_0 \left(1 - e^{-\frac{t}{	au}}ight)$$

Here, $I_0$ is the maximum, steady-state current, and $	au$ (tau) is the time constant of the circuit.

### The Time Constant: A Measure of Sluggishness

The time constant $	au$ dictates how quickly the current reaches its maximum value. It is defined as:
$$	au = \frac{L}{R}$$

A larger inductance $L$ means a stronger back EMF, making the circuit more sluggish (larger $	au$). Conversely, a larger resistance $R$ means the final steady-state current $I_0 = \frac{E}{R}$ is smaller. Because the target current is lower, the circuit reaches $63.2\%$ of that target in a shorter amount of time, hence a smaller $	au$.

### Solving the Problem at Hand

Let's apply these beautiful concepts to our specific problem. We are given:
- Inductance, $L = 10	ext{ H}$
- Resistance, $R = 5\,\Omega$
- Battery Voltage, $E = 5	ext{ V}$
- Time elapsed, $t = 2	ext{ s}$

**Step 1: Find the maximum current ($I_0$)**
If we wait an infinitely long time, the current stops changing ($\frac{dI}{dt} = 0$). The inductor acts just like a regular piece of wire. The circuit is then purely resistive.
$$I_0 = \frac{E}{R} = \frac{5}{5} = 1	ext{ A}$$

**Step 2: Find the time constant ($	au$)**
$$	au = \frac{L}{R} = \frac{10}{5} = 2	ext{ s}$$

**Step 3: Calculate the instantaneous current**
We want the current exactly $2	ext{ s}$ after the switch is closed. We plug our values into the growth equation:
$$I(2) = 1 \cdot \left(1 - e^{-\frac{2}{2}}ight)$$
$$I(2) = 1 - e^{-1}	ext{ A}$$

### The Beauty of the Result

Our final answer is $1 - e^{-1}$ amperes. Since $e \approx 2.718$, $e^{-1} \approx 0.368$. This means that after exactly one time constant ($	au = 2	ext{ s}$), the current has reached $1 - 0.368 = 0.632$, or $63.2\%$ of its maximum possible value. This is a universal truth for all exponential growth processes in physics, beautifully demonstrated by this simple circuit!

Animated Solution for Physics - Electromagnetic Induction: An ideal coil of $10 H$ is connected in series with a resistance of $5 Ω$ and a battery of $5 V$ . After $2 s$ , the connection is made, the current flowing (in ampere) in the circuit is

Select Answer:

Visualized Solution

$L-R Circuit Setup$

$Current Growth Equation$

$Maximum Current (I_{0})$

$Time Constant (τ)$

$Substituting Values$

$Final Calculation$

$The Way Forward$

The Sigma Insight: Alternating Current (AC) and Voltage

The Stubborn Nature of Inductors

Setting Up the Differential Equation

The Time Constant

Solving the Problem at Hand

The Beauty of the Result

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Visualized Solution

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

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Maximum Current (I0​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Time Constant (τ)

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

.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