(a) The current through a wire is a steady 2.5 amps. How much current passes through it between t = 0 seconds and t = 4 × 10-4 seconds?
(b) The current through a wire is given by I(t) = I0e -at, where I0 = 2.5 amps and a = 6 × 103 s-1. How much electric charge passes through the wire between t = 0 seconds and t = 4 × 10-4 seconds?
a) To find the amount of current passing through the wire between t = 0 seconds and t = 4 × 10^-4 seconds, we need to calculate the total charge.
The formula to calculate the charge is Q = I * t, where Q is the charge, I is the current, and t is the time.
Given that the current is a steady 2.5 amps, we can substitute the values into the formula:
Q = 2.5 amps * (4 × 10^-4 seconds - 0 seconds)
Q = 2.5 amps * (4 × 10^-4 seconds)
Q = 10^-3 coulombs
Therefore, the amount of current passing through the wire between t = 0 seconds and t = 4 × 10^-4 seconds is 10^-3 coulombs.
b) In this case, we are given the current as a function of time: I(t) = I0e^(-at). We need to calculate the total charge passing through the wire between t = 0 seconds and t = 4 × 10^-4 seconds.
To find the total charge, we use the formula Q = ∫ I(t) dt from t = 0 to t = 4 × 10^-4 seconds.
Substituting the given values into the formula:
Q = ∫ (2.5 amps * e^(-6 × 10^3 s^-1 t)) dt from 0 to 4 × 10^-4 seconds
Integrating this equation will give us the total charge passing through the wire within the given time interval.
Using integration techniques, we can solve the integral and find the value of Q.
Note: Since this problem involves calculus, it's recommended to use an appropriate tool or software to calculate the definite integral.