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Question #42990

Difficulty: 8/10 | 🟢 Free

The Question

A real solenoid of inductance $L$ (in $\text{H}$ ) and internal resistance $r$ (in $\Omega$ ) is connected in series with a variable load resistance $R$ (in $\Omega$ ) and an ideal DC voltage source of electromotive force $V$ (in $\text{V}$ ). The circuit is closed at $t=0$ and allowed to reach a steady state. Once the steady state is reached, the voltage source is instantaneously bypassed by a short circuit, allowing the current to decay through the resistors. Let $E_R$ (in $\text{J}$ ) be the total energy dissipated in the load resistance $R$ during the entire decay process. Which of the following gives the value of $R$ that maximizes $E_R$ , and the corresponding maximum energy $(E_R)_{\max}$ ?

Options

A.
$R = \frac{r}{2}$ , $(E_R)_{\max} = \frac{2 V^2 L}{27 r^2}$
B.
$R = r$ , $(E_R)_{\max} = \frac{V^2 L}{16 r^2}$
C.
$R = \frac{r}{3}$ , $(E_R)_{\max} = \frac{9 V^2 L}{128 r^2}$
D.
$R = 2r$ , $(E_R)_{\max} = \frac{V^2 L}{27 r^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Self-Inductance of Solenoid|Transient Current in RL Circuit|Maximum Power Transfer

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between steady-state power optimization and transient energy optimization.

A lethal and highly common trap is blindly applying the Maximum Power Transfer Theorem ( $R = r$ ), which maximizes the instantaneous power delivered to a load from a constant voltage source.

However, during the transient decay phase, the energy source is the inductor, whose initial stored energy $\frac{1}{2}LI_0^2$ depends inversely on $(R+r)^2$ .

The total energy dissipated in the load is the product of this stored energy and the energy-sharing ratio $\frac{R}{R+r}$ .

Maximizing this product requires differentiating the transient energy integral with respect to $R$ , yielding $R = r/2$ .

Students who fail to account for the dependence of the initial inductive energy barrier on the load resistance will incorrectly choose the MPTT result.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Self-Inductance of Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free