Ch 28: Fundamentals of Circuits

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 28: Fundamentals of Circuits

문제 69

28장, 문제 69

A 150 μF defibrillator capacitor is charged to 1500 V. When fired through a patient’s chest, it loses 95% of its charge in 40 ms. What is the resistance of the patient’s chest?

검증된 단계별 안내

Step 1: Recognize that the capacitor discharges through the patient's chest, which can be modeled as an RC circuit. The voltage across the capacitor during discharge follows the equation:

V_{t} = V_{0} e^{- \frac{t}{τ}}

, where

τ

is the time constant of the circuit.

Step 2: The time constant

τ

is given by the formula:

τ = R C

, where

R

is the resistance and

C

is the capacitance.

Step 3: Use the information that the capacitor loses 95% of its charge in 40 ms. This means the voltage drops to 5% of its initial value. Substitute this into the discharge equation:

V_{t} = V_{0} e^{- \frac{t}{τ}}

, where

V_{t} = 0.05 V_{0}

and

t = 40 ms

Step 4: Solve for the time constant

τ

using the equation:

e^{- \frac{t}{τ}} = 0.05

. Take the natural logarithm of both sides to isolate

\frac{t}{τ}

- \frac{t}{τ} = \ln (0.05)

. Rearrange to find

τ

τ = \frac{t}{-}

Step 5: Once

τ

is calculated, use the formula

τ = R C

to solve for the resistance

R

. Substitute the given capacitance

C = 150 μ F

and the calculated value of

τ

into the equation to find

R

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주요 개념

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

Capacitance

Capacitance is the ability of a capacitor to store electrical charge per unit voltage. It is measured in farads (F), and in this case, the defibrillator capacitor has a capacitance of 150 μF (microfarads). The charge (Q) stored in a capacitor can be calculated using the formula Q = C × V, where C is capacitance and V is voltage.

Exponential Decay of Charge

When a capacitor discharges through a resistor, the charge decreases exponentially over time. The relationship is described by the equation Q(t) = Q0 * e^(-t/RC), where Q0 is the initial charge, R is resistance, and C is capacitance. In this scenario, the capacitor loses 95% of its charge in 40 ms, which can be used to find the resistance of the patient's chest.

Ohm's Law

Ohm's Law relates voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = I × R. In the context of the defibrillator, understanding this relationship is crucial for calculating the resistance of the patient's chest based on the current flowing through it as the capacitor discharges. This law helps in determining how the resistance affects the rate of charge loss.