Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

문제 35

30장, 문제 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?

검증된 단계별 안내

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 = I_{0} e^{- \frac{t}{τ}}

, where

I

is the current at time

t

I_{0}

is the initial current, and

τ

is the time constant of the circuit.

Substitute the condition given in the problem: the current is

\frac{1}{2} I_{0}

at some time

t

. This gives the equation:

\frac{1}{2} I_{0} = I_{0} e^{- \frac{t}{τ}}

Simplify the equation by dividing both sides by

I_{0}

, resulting in:

\frac{1}{2} = e^{- \frac{t}{τ}}

. Take the natural logarithm of both sides to solve for

t

- \frac{t}{τ} = \ln (\frac{1}{2})

Rearrange the equation to isolate

t

t = - τ \ln (\frac{1}{2})

. 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.