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Diagram

$4.8 \Omega$

$4 \Omega$

$8.4 \Omega$

$40 \Omega$

Correct Answer

⚙️ TE • Technical Faraday's Law AnalysisTechnical AbilityElectrical Circuits

Option A

$8 \Omega$

Quick Summary:

To determine the impedance of the coil, we first calculate the induced electromotive force (EMF) using Faraday's Law of Electromagnetic Induction, which states that the induced EMF is the negative rate of change of magnetic flux ( $e = -\frac{d\phi}{dt}$ ). Since the resistance of the coil is $R = \frac{e}{i}$ , we calculate the magnitude of the EMF at $t = 2$ seconds and divide by the given current $i = 5\text{ A}$ .

⚙️TETechnical SolutionFaraday's Law Analysis

💡 Explanation

🔢 Key Formulas

$e = \frac{d\phi}{dt}$ — Induced EMF from magnetic flux

$Z = \frac{e}{i}$ — Impedance derived from Ohm's law

⚙️ Working Principle

According to Faraday's Law, the induced EMF magnitude is $|e| = |\frac{d\phi}{dt}|$ . Given $\phi(t) = 5t^{2} + 4t + 10$ , the derivative is $\frac{d\phi}{dt} = 10t + 4$ . At $t = 2\text{ s}$ , $|e| = 10(2) + 4 = 24\text{ V}$ . Using Ohm's Law, the impedance (in this case, resistance as the circuit is treated as purely resistive) is $Z = \frac{e}{i} = \frac{24\text{ V}}{5\text{ A}} = 4.8\text{ }\Omega$ .

📌 Key Points

▸
Faraday's Law relates magnetic flux variation to induced EMF.
▸
The impedance $Z$ for a purely inductive or resistive coil at a specific instant is calculated using instantaneous values of voltage and current.
▸
The constant term in the flux equation ( $10\text{ Wb}$ ) becomes zero upon differentiation.

✅ Advantages

▸
Simple application of calculus to solve electromagnetic problems.
▸
Direct calculation based on given time-dependent parameters.

❌ Disadvantages / Limitations

▸
Assumes the coil behavior is linear and that instantaneous impedance equals DC resistance at the given frequency/time.
▸
Requires knowledge of derivatives.

🛠️ Applications / Uses

▸
Design of electromagnetic sensors.
▸
Analysis of transient response in R-L circuits.

📄 Additional Information

▸
The derivative of $\phi = 5t^{2} + 4t + 10$ with respect to time is $10t + 4$ .
▸
At $t = 2$ , $e = 24\text{ V}$ .
▸
Given $i = 5\text{ A}$ , $R = \frac{24}{5} = 4.8\text{ }\Omega$ .

📊 Diagram / Illustration

✅

A is correct — The calculated impedance based on induced EMF and current is $4.8\text{ }\Omega$ .

Core Concepts Used

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Faraday's Law of Induction Instantaneous Current and Voltage Impedance Definition

💡 EXAM TIP

Always remember that in problems involving time-varying magnetic flux, the derivative represents the instantaneous EMF; never forget to differentiate the constant term, which disappears.

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