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Ch 27: Current and Resistance
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 27: Current and ResistanceProblema 59b
Capitolo 27, Problema 59b

The total amount of charge that has entered a wire at time t is given by the expression Q=(20C)(1−e−t2.0s)Q=\(\left\)(20C\(\right\))\(\left\)(1-e^{-\(\frac{t}{2.0s}\)}\(\right\)), where t is in seconds and t≥0. What is the maximum value of the current?

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The current, I, is the rate of change of charge, Q, with respect to time, t. Mathematically, this is expressed as I = dQ/dt. Start by differentiating the given expression for Q with respect to t.
The given expression for Q is Q = (20C)(1 - e^(-t/2.0s)). Use the chain rule to differentiate this expression. The derivative of a constant (20C) multiplied by a function is the constant multiplied by the derivative of the function.
The derivative of (1 - e^(-t/2.0s)) is 0 - (-1)(e^(-t/2.0s))(1/2.0s), where the factor (1/2.0s) comes from the chain rule applied to the exponent -t/2.0s. Simplify this derivative.
Combine the results to find the expression for the current: I = (20C)(e^(-t/2.0s))(1/2.0s). Simplify further to get I = (10C/s)(e^(-t/2.0s)).
To find the maximum value of the current, note that the exponential term e^(-t/2.0s) decreases as t increases. The maximum value of the current occurs at t = 0. Substitute t = 0 into the expression for I to determine the maximum current.

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Concetti chiave

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Charge (Q)

Charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. In this context, the total charge Q that has entered the wire is expressed as a function of time t, indicating how charge accumulates over time. The equation provided shows that charge approaches a maximum value as time increases, reflecting the behavior of a charging capacitor.
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Conservation of Charge

Current (I)

Current is defined as the rate of flow of electric charge through a conductor, typically measured in amperes (A). It can be calculated as the derivative of charge with respect to time (I = dQ/dt). In this scenario, understanding how to derive the current from the given charge function is essential to determine its maximum value as the system reaches steady state.
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Intro to Current

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the given equation, the term involving 'e' represents this decay, indicating how the charge approaches its maximum value over time. This concept is crucial for understanding how quickly the current reaches its peak as the charge accumulates in the wire.
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Amplitude Decay in an LRC Circuit
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