Exponential decay is a mathematical model that describes the gradual reduction of a quantity over time. It is characterized by an initial value that decreases at a rate proportional to its current value, resulting in a smooth, continuous decline.
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Exponential decay is often represented by the equation $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial value, $k$ is the decay constant, and $t$ is the time.
In damped harmonic motion, the amplitude of the oscillations decreases exponentially over time due to the dissipation of energy.
In DC circuits with resistors and capacitors, the voltage across the capacitor decreases exponentially as it discharges through the resistor.
The half-life of a radioactive substance is the time it takes for the activity to decrease to half of its initial value, which is a form of exponential decay.
The time constant in exponential decay is the time it takes for a quantity to decrease to 1/e (approximately 37%) of its initial value.
Review Questions
Explain how exponential decay is observed in damped harmonic motion and describe the key parameters that govern the decay process.
In damped harmonic motion, the amplitude of the oscillations decreases exponentially over time due to the dissipation of energy. This can be described by the equation $A(t) = A_0 e^{- extbackslashgamma t}$, where $A_0$ is the initial amplitude, extbackslashgamma is the damping coefficient, and $t$ is the time. The damping coefficient extbackslashgamma determines the rate of exponential decay, with higher values leading to a faster decrease in amplitude. The time constant, which is the time it takes for the amplitude to decrease to 1/e (approximately 37%) of its initial value, is given by $ au = 1/ extbackslashgamma$. Understanding the parameters that govern exponential decay in damped harmonic motion is crucial for analyzing the behavior of oscillating systems.
Describe the role of exponential decay in the behavior of DC circuits containing resistors and capacitors, and explain how the time constant of the circuit affects the discharge process.
In DC circuits with resistors and capacitors, the voltage across the capacitor decreases exponentially as it discharges through the resistor. This exponential decay is described by the equation $V(t) = V_0 e^{-t/RC}$, where $V_0$ is the initial voltage, $R$ is the resistance, $C$ is the capacitance, and $t$ is the time. The time constant of the circuit, given by $ au = RC$, determines the rate of the exponential decay. A larger time constant means a slower discharge, as the capacitor takes longer to lose a significant portion of its stored charge. Understanding the role of the time constant in exponential decay is crucial for analyzing the transient behavior of RC circuits and predicting the voltage changes over time.
Explain the connection between exponential decay and the concept of half-life, and discuss how this relationship is applied in the study of radioactive decay and the measurement of radioactivity.
The half-life of a radioactive substance is the time it takes for the activity to decrease to half of its initial value, which is a form of exponential decay. The relationship between half-life and exponential decay is given by the equation $N(t) = N_0 e^{- extbackslashlambda t}$, where $N_0$ is the initial activity, extbackslashlambda is the decay constant, and $t$ is the time. The half-life is related to the decay constant by the equation $t_{1/2} = extbackslashln(2) / extbackslashlambda$. This understanding of exponential decay and half-life is fundamental to the study of radioactive decay and the measurement of radioactivity, as it allows for the prediction of the remaining activity of a radioactive substance over time. The concept of half-life is widely used in fields such as nuclear physics, radiochemistry, and medical imaging, where the exponential decay of radioactive materials is a critical factor.
Related terms
Half-Life: The time it takes for a quantity to decrease to half of its initial value, a key concept in exponential decay.