Before You Begin

Question #42805

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $+q$ is attached to the lower end of a massless spring of spring constant $k$ . The upper end of the spring is fixed to a rigid support. The system is initially at rest in its vertical equilibrium position under gravity. At $t = 0$ , a time-varying vertical electric field $E(t) = E_0 \sin(\omega t)$ is switched on, pointing downwards. Given that the angular frequency of the electric field is $\omega = \frac{1}{2} \sqrt{\frac{k}{m}}$ , what is the total mechanical energy of the oscillator (sum of kinetic, elastic potential, and gravitational potential energies) at time $t = \frac{\pi}{\omega}$ , relative to its initial equilibrium state at $t=0$ ?

Options

A.
$\frac{8 q^2 E_0^2}{9 k}$
B.
$0$
C.
$\frac{q^2 E_0^2}{2 k}$
D.
$\frac{2 q^2 E_0^2}{3 k}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Work against Conservative Force|Vertical Oscillations|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of energy conservation in time-varying fields.

Students often incorrectly assume that because the electrostatic force is conservative at any given instant, they can define a time-dependent potential energy $U_E(t) = -qE(t)y$ and conserve the total energy $E_{mech} + U_E = \text{constant}$ .

Since $E(\pi/\omega) = 0$ and $y(\pi/\omega) = 0$ , this flawed logic leads to the conclusion that $\Delta E_{mech} = 0$ .

However, the Hamiltonian of the system explicitly depends on time, meaning total mechanical energy is not conserved; the electric field does net work on the oscillator over the half-cycle, which must be calculated via the Work-Energy Theorem or by explicitly solving the non-autonomous differential equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Work against Conservative Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free