A voltage source of V(t) = (10t 3 -5t + 10 ) Volt is applied a... | Technical Ability MCQ | Examoogle

Technical AbilityElectrical Circuit Theory

A voltage source of V(t) = (10t 3 -5t + 10 ) Volt is applied across a 10F capacitor, the current through the capacitor at t = 2 sec is _________.

1100 A

1150 A

1160 A

1000 A

Correct Answer

⚙️ TE • Technical Concept & PrincipleTechnical AbilityElectrical Circuit Theory

Option B

1150 A

Quick Summary:

The current $i(t)$ through a capacitor is determined by the rate of change of voltage across its terminals, expressed by the relationship $i(t) = C \cdot \frac{dv(t)}{dt}$ . Given the voltage function $v(t) = 10t^{3} - 5t + 10$ and capacitance $C = 10\text{F}$ , we differentiate the voltage with respect to time to obtain the current expression.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$i(t) = C \cdot \frac{dv(t)}{dt}$ — The time-domain current-voltage relationship for a capacitor

$\frac{d}{dt}(at^{n}) = n \cdot at^{n-1}$ — The power rule for differentiation

⚙️ Working Principle

A capacitor stores energy in an electric field. The fundamental constitutive relation $i = C \frac{dv}{dt}$ defines the current as directly proportional to the time derivative of the applied voltage. By differentiating $v(t) = 10t^{3} - 5t + 10$ , we get $v'(t) = 30t^{2} - 5$ . Evaluating this at $t = 2\text{s}$ gives $30(4) - 5 = 115\text{V/s}$ . Multiplying by $C = 10\text{F}$ results in $1150\text{A}$ .

📌 Key Points

▸
The capacitor current is zero if the voltage across it remains constant over time.
▸
A discontinuous change in voltage across a capacitor would require an infinite current, which is physically impossible in practical circuits.
▸
Units must be consistent: Voltage in Volts (V), Time in seconds (s), and Capacitance in Farads (F) to yield current in Amperes (A).

✅ Advantages

▸
Provides a direct method to relate terminal characteristics of dynamic passive elements.
▸
Essential for analyzing transient responses in RLC circuits.

❌ Disadvantages / Limitations

▸
Requires knowledge of calculus for time-varying inputs.
▸
Ideal model assumes no leakage resistance or parasitic inductance.

🛠️ Applications / Uses

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Filter design in power electronics.
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Signal processing and coupling circuits.

📄 Additional Information

▸
For $t = 2\text{s}$ , the derivative $\frac{dv}{dt} = 30t^{2} - 5 = 30(2)^{2} - 5 = 120 - 5 = 115\text{V/s}$ .
▸
Multiplying the derivative by the capacitance $C = 10\text{F}$ gives $10 \times 115 = 1150\text{A}$ .
▸
Other options are mathematically incorrect results obtained from miscalculating the derivative or exponent values.

📊 Diagram / Illustration

✅

B is correct — Using the derivative $i = C \frac{dv}{dt}$ , the calculated current at $t = 2\text{s}$ is $1150\text{A}$ .

Core Concepts Used

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Capacitor V-I Characteristic Differentiation of polynomials Transient circuit analysis

💡 EXAM TIP

Always check for units; if the capacitance was given in microfarads ( $\mu\text{F}$ ), the current result would be scaled by $10^{-6}$ .

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