Your cookie choices
Essential cookies keep your basket and sign-in working. Optional cookies help us understand visits and measure ads. Privacy details.
Essential cookies keep your basket and sign-in working. Optional cookies help us understand visits and measure ads. Privacy details.
How quantities decrease over time (radioactivity, capacitors)
AI-assisted explanation. It may contain errors; use a textbook or original source to check important details.
This equation says: whatever quantity you’re tracking (call it u) fades away smoothly and predictably over time, dropping by the same fraction in equal time steps. Think of u(t) as the “amount left” at time t—charge left on a capacitor, undecayed radioactive atoms, the leftover temperature difference between a hot drink and the room, or the remaining intensity of a signal after losses. The special thing about exponential decay is that it doesn’t lose a fixed amount per second; it loses a fixed percentage per second. Big at first, so it falls fast; smaller later, so it falls more gently. The characters in the story: - u(t): the remaining fraction/amount at time t (often dimensionless if u(0)=1). - e: Euler’s number (~2.718), the “natural” base that appears whenever change is proportional to what’s currently there. - t: time. - τ (tau): the time constant, the clock that sets the pace of fading. What τ means physically is beautifully concrete: after one time constant, t = τ, u(τ) = ≈ 0.368. So in one τ, you fall to about 37% of what you started with (and you’ve lost about 63%). After 2τ you’re at ≈ 13.5%, after 3τ about 5%, and so on. τ is the system’s built-in “forgetting time.”
Exponential decay grew out of a 17th–18th century revolution: the invention of calculus and the realization that many natural processes are governed by a simple rule—the rate of change is proportional to the current amount. Mathematicians like Jacob Bernoulli studied the exponential function while investigating compounding and continuous change, and calculus provided the language to express “proportional to itself” precisely. In the 19th century, this idea became a workhorse of physics through differential equations describing cooling, damping, and electrical circuits. In the early 20th century, exponential decay became famously tied to radioactivity. Experiments showed that a sample’s activity decreases in a way that depends only on how many unstable nuclei remain—leading to the exponential law and concepts like half-life. Across disciplines, the same mathematical shape kept reappearing, signaling a universal mechanism: a constant probability per unit time for each individual ‘unit’ to disappear (atom decays, charges leak, excited states relax).
Pioneered by: There isn’t a single discoverer of u(t)=; it is the standard solution to a first-order linear differential equation. Historically, the exponential function and its connection to proportional change were developed by early calculus pioneers (notably Jacob Bernoulli and later Leonhard Euler). In specific physical contexts, the exponential decay law was prominently established in radioactivity by Ernest Rutherford and Frederick Soddy (early 1900s), and in circuits it follows from Kirchhoff’s laws combined with Ohm’s law and capacitance/inductance.
Physics-inspired apparel
