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Ch 30: Electromagnetic Induction
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 30: Electromagnetic InductionProblema 35
Capitolo 30, Problema 35

At t = 0 s, the current in the circuit in FIGURE EX30.35 is I0. At what time in μs is the current (1/2)I0?

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1
Identify the type of circuit in the problem. Based on the context, this is likely an RL (resistor-inductor) circuit, where the current decays exponentially over time after the circuit is disconnected from a power source.
Write the equation for the current in an RL circuit as a function of time: I=I0⁢e⁢−tτ, where I is the current at time t, I0 is the initial current, and τ is the time constant of the circuit.
Substitute the condition given in the problem: the current is 12I0 at some time t. This gives the equation: 12I0=I0⁢e⁢−tτ.
Simplify the equation by dividing both sides by I0, resulting in: 12=e⁢−tτ. Take the natural logarithm of both sides to solve for t: −tτ=ln(12).
Rearrange the equation to isolate t: t=−τ⁢ln(12). Substitute the value of the time constant τ (if provided in the problem or determined from the circuit parameters) to calculate the time t in microseconds.

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Current in a Circuit

Current is the flow of electric charge in a circuit, measured in amperes (A). In this context, I₀ represents the initial current at time t=0 s. Understanding how current changes over time in response to circuit components, such as resistors and capacitors, is crucial for solving the problem.
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03:45
Current in a Parallel RC AC Circuit

Exponential Decay

In circuits involving resistors and capacitors, the current often decreases exponentially over time due to the discharge of the capacitor. The mathematical representation of this decay can be expressed as I(t) = I₀ * e^(-t/τ), where τ is the time constant. Recognizing this behavior is essential for determining when the current reaches half of its initial value.
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Amplitude Decay in an LRC Circuit

Time Constant (τ)

The time constant τ is a key parameter in RC circuits, defined as τ = R*C, where R is resistance and C is capacitance. It indicates the time it takes for the current to decrease to approximately 37% of its initial value. This concept helps in calculating the time required for the current to reach specific fractions of its initial value, such as 1/2 I₀.
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Phase Constant of a Wave Function
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