What is the formula for decay rate in physics?

The formula for decay rate in physics is \( N(t) = N_0 e^{-\lambda t} \).

In more detail, the decay rate formula describes how the number of undecayed particles or nuclei in a sample decreases over time. Here, \( N(t) \) represents the number of particles remaining at time \( t \), \( N_0 \) is the initial number of particles at the start (when \( t = 0 \)), \( \lambda \) is the decay constant, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

The decay constant \( \lambda \) is a measure of how quickly the substance decays. A larger \( \lambda \) means the substance decays more quickly. The term \( e^{-\lambda t} \) shows the exponential nature of the decay process, meaning the number of particles decreases rapidly at first and then more slowly over time.

To understand this better, imagine you have a sample of a radioactive substance. If you start with 1000 particles and the decay constant \( \lambda \) is 0.1 per year, you can use the formula to find out how many particles remain after a certain number of years. For example, after 5 years, the number of particles left would be \( 1000 \times e^{-0.1 \times 5} \).

This formula is crucial in fields like nuclear physics, archaeology (for carbon dating), and medicine (for understanding the behaviour of radioactive tracers). It helps scientists and researchers predict how long a substance will remain active or how quickly it will diminish.