Consider the camera flash in Problem 96 . If the flash...

Consider the camera flash in Problem $96 .$ If the flash really discharges according to Eq. $(18-48),$ then it takes an infinite amount of time to discharge. When Problem 96 assumes that the capacitor discharges in $4.0 \times 10^{-3} \mathrm{s},$ we mean that the capacitor has almost no charge stored on it after that amount of time. Suppose that after $4.0 \times 10^{-3} \mathrm{s}$ the capacitor has only $1.0 \%$ of the original charge still on it. (a) What is the time constant of this $R C$ circuit? (b) What is the resistance of the flashbulb in this case? (c) What is the maximum power dissipated in the flashbulb?

Consider the camera flash in Problem $96 .$ If the flash really discharges according to Eq. $(18-48),$ then it takes an infinite amount of time to discharge. When Problem 96 assumes that the capacitor discharges in $4.0 \times 10^{-3} \mathrm{s},$ we mean that the capacitor has almost no charge stored on it after that amount of time. Suppose that after $4.0 \times 10^{-3} \mathrm{s}$ the capacitor has only $1.0 \%$ of the original charge still on it. (a) What is the time constant of this $R C$ circuit? (b) What is the resistance of the flashbulb in this case?
(c) What is the maximum power dissipated in the flashbulb?

Key Concepts

Maximum Power Dissipation in a Resistive Load

The maximum power dissipation in a resistor connected to a capacitor occurs typically at the initial moment when the voltage or current is at its highest. It can be calculated using the instantaneous power formulas, such as P = V^2/R or P = I^2R, and understanding the initial conditions of the circuit. This concept is crucial when designing circuits that must handle peak power levels safely.

Determination of the RC Time Constant from Decay Percentage

This involves using the exponential decay formula to relate the remaining charge (expressed as a percentage of the initial charge) after a certain period to the time constant ?. By setting up the equation Q(t)/Q0 = exp(?t/?) and solving for ? given the percentage decay and time interval, one can determine the characteristic time scale of the circuit's response.

Calculation of Resistance from Time Constant

Once the time constant and capacitance are known, the resistance in the circuit can be calculated using the relation ? = RC. This step is fundamental in reverse-engineering the circuit’s parameters from experimental or prescribed decay characteristics.

Exponential Decay in RC Circuits

This concept describes the way in which a capacitor discharges through a resistor following an exponential law. The voltage or charge on the capacitor decreases as an exponential function of time, mathematically expressed as Q(t) = Q0 exp(?t/?), where Q0 is the initial charge, t is time, and ? is the time constant. This model is essential for understanding how quickly the energy stored in the capacitor is released.

Time Constant (?)

The time constant in an RC circuit, denoted by ?, is the product of the resistance (R) and the capacitance (C) of the circuit (? = RC). It represents the time required for the charge (or voltage) on the capacitor to decrease to about 36.8% of its initial value. Knowing ? is key to predicting the dynamic behavior of the circuit during charging or discharging.

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