Q.29 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 29 Solution

Topic: Electromagnetic Waves · Official NTA Booklet Code M1

Question

A capacitor of capacitance 'C', is connected across an ac source of voltage V, given by V = V₀ sin ωt. The displacement current between the plates of the capacitor, would then be given by:

✓

Verified Against NTA Final Answer Key

Option (1) : I_d = V_0 ω C cos(ωt)

📚 Electromagnetic Waves Easy

📋 Given Parameters & Problem Setup

Capacitance C

Applied AC Voltage V = V_0 sin(ωt)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Displacement Current = Conduction Current: I_d = dq/dt

⏱️ Save ~20 seconds

By Maxwell's continuity theorem, displacement current inside the dielectric gap equals the conduction current in the connecting wires: $I_d = I_c = \frac{dq}{dt} = C \frac{dV}{dt} = C \frac{d}{dt}[V_0 \sin(\omega t)] = V_0 \omega C \cos(\omega t)$ ! Instant 10-second derivative matching Option (1)!

1 Charge on capacitor plates at any instant:

q(t) = C V(t) = C V_0 \sin(\omega t)

2 By Maxwell-Ampere law, displacement current between plates equals conduction current:

I_d = \frac{dq}{dt}

3 Differentiate with respect to time:

\frac{d}{dt}[V_0 \sin(\omega t)] = V_0 \omega \cos(\omega t)

4 Thus,

I_d = V_0 \omega C \cos(\omega t)

5 Match Option (1).

🛡️

Why You Can 100% Trust This Shortcut: Canonical Maxwell-Ampere circuital law and charge-current continuity equation.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Definition of Displacement Current

Maxwell defined displacement current $I_d$ in terms of the rate of change of electric flux $\Phi_E$ between the capacitor plates:

I_d = \varepsilon_0 \frac{d\Phi_E}{dt}

Step 2

Expressing Electric Flux in Terms of Charge and Voltage

For a parallel plate capacitor of area $A$ and separation $d$ , the uniform electric field is $E = V/d$ . The electric flux is $\Phi_E = E A = (V/d)A = V (A/d)$ :

\varepsilon_0 \Phi_E = \varepsilon_0 \left(\frac{A}{d}\right) V = C V = q

Step 3

Evaluating the Time Derivative

Substitute $V(t) = V_0 \sin(\omega t)$ into the displacement current expression:

I_d = \frac{d}{dt}(C V) = C \frac{dV}{dt} = C \frac{d}{dt}\left[V_0 \sin(\omega t)\right] = V_0 \omega C \cos(\omega t)

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 1

✓ 100% Correct

Verdict & Scientific Reason:

Exact mathematical derivative: $I_d = C \frac{dV}{dt} = V_0 \omega C \cos(\omega t)$ .

💡

Why this is airtight: None; textbook calculus and electromagnetic derivation.

2 2

✗ Incorrect

Verdict & Scientific Reason:

Divides by $\omega C$ instead of multiplying (reactance is $X_C = 1/\omega C$ , so current amplitude is $V_0 / X_C = V_0 \omega C$ ).

⚠️

Student Fallacy (Why candidates pick this trap): Writing

I_0 = V_0 X_C

with

X_C = 1/\omega C

placed in the denominator incorrectly.

3 3

✗ Incorrect

Verdict & Scientific Reason:

Contains inverted impedance and fails to take the time derivative of sine into cosine.

⚠️

Student Fallacy (Why candidates pick this trap): Forgetting that current leads voltage by

90^\circ

(cosine function).

4 4

✗ Incorrect

Verdict & Scientific Reason:

Fails to differentiate $\sin(\omega t)$ to $\cos(\omega t)$ (phase is off by $90^\circ$ ).

⚠️

Student Fallacy (Why candidates pick this trap): Neglecting the derivative of the sinusoidal voltage waveform.

⚠️

NTA Trap Alert: Sine vs. Cosine Differentiation Trap

Students who remember the current amplitude $I_0 = V_0 \omega C$ sometimes pick Option (4) $V_0 \omega C \sin(\omega t)$ , forgetting that differentiation of $\sin(\omega t)$ produces $\cos(\omega t)$ (representing the $90^\circ$ phase lead of capacitor current).

🛡️ How to avoid the -1 mark: Differentiate with respect to time:

\frac{d}{dt}[\sin(\omega t)] = \omega \cos(\omega t)

📖

Core NEET Syllabus Standard

NCERT Physics · Electromagnetic Waves Standard Benchmark

Core Scientific Principle Tested:

"Displacement current between plates is identical to the conduction current in external leads: $I_d = \varepsilon_0 \frac{d\Phi_E}{dt} = C \frac{dV}{dt}$ ."

Syllabus Focus: Displacement Current

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 29 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Displacement Current = Conduction Current: I_d = dq/dt

Step-by-Step Rigorous Workflow & Mathematical Proof

Definition of Displacement Current

Expressing Electric Flux in Terms of Charge and Voltage

Evaluating the Time Derivative

360° Option Elimination Engine Key Feature

NTA Trap Alert: Sine vs. Cosine Differentiation Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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EM Wave in Dielectric Medium: Dual-Key Question Audit (Both Options 2 and 3 Accepted by NTA)

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