The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is $I=\left(0.75A\right)e^{-\frac{t}{6h}}$. What is the total number of electrons transported from the positive electrode to the negative electrode by the charge escalator from the time the battery is first used until it is completely dead?

Verified step by step guidance

1. The total number of electrons transported can be determined by first calculating the total charge that flows through the circuit over time. The total charge is the integral of the current over time, since current is the rate of flow of charge. Start by writing the expression for the current: I(t) = (0.75 A) * e^(-t / (6 h)).
2. The total charge Q is given by the integral of the current I(t) with respect to time t, from t = 0 to t = ∞. Mathematically, Q = ∫ I(t) dt = ∫ (0.75 A) * e^(-t / (6 h)) dt, where the limits of integration are from 0 to ∞.
3. To solve the integral, note that the exponential function e^(-t / (6 h)) has a standard integral form. Rewrite the integral as Q = 0.75 A * ∫ e^(-t / (6 h)) dt. Use the substitution u = t / (6 h), which implies du = dt / (6 h). This simplifies the integral to Q = 0.75 A * (6 h) * ∫ e^(-u) du.
4. The integral of e^(-u) with respect to u is simply -e^(-u). Apply the limits of integration (from u = 0 to u = ∞) to find the total charge Q. After evaluating the definite integral, you will have the total charge transported by the battery.
5. Finally, to find the total number of electrons transported, use the relationship between charge and the elementary charge of an electron: N = Q / e, where e = 1.6 × 10^(-19) C is the charge of a single electron. Divide the total charge Q by e to calculate the total number of electrons transported.

Key Concepts

Current and Exponential Decay

Current, measured in amperes (A), represents the flow of electric charge. In this scenario, the current decreases exponentially over time, as described by the function I(t) = 0.75 A * e^(-t/6h). This indicates that as time progresses, the current diminishes, reflecting the battery's depletion.

Charge and Electrons

Electric charge is carried by electrons, with one coulomb (C) equivalent to approximately 6.242 x 10^18 electrons. The total charge transported can be calculated by integrating the current over time, which gives the total charge in coulombs. This charge can then be converted to the number of electrons by dividing by the charge of a single electron.

Integration in Physics

Integration is a fundamental mathematical tool used to calculate the total quantity from a rate of change. In this context, integrating the current function I(t) over the time interval from 0 to the time when the battery is dead allows us to find the total charge transferred. This process is essential for determining how much charge has moved through the circuit during the battery's operation.