The current in a wire varies with time according to the relationship I = 55 A - (0.65 A/s^2)t^2. (b) What constant current would transport the same charge in the same time interval?

Verified step by step guidance

1. Identify the expression for the current as a function of time, which is given by I(t) = 55 A - (0.65 A/s^2)t^2.
2. Recognize that the total charge transported by the current over a time interval can be found by integrating the current function with respect to time. The formula to use is Q = \int I(t) dt, where Q is the total charge.
3. Set up the integral for the current function from t = 0 to t = T, where T is the total time interval. The integral becomes Q = \int_0^T (55 - 0.65t^2) dt.
4. Solve the integral to find the expression for Q, the total charge transported over the time interval T.
5. To find the constant current I_const that would transport the same charge Q in the same time interval T, use the formula I_const = Q / T. This gives you the constant current equivalent to the varying current over the same period.

Key Concepts

Current (I)

Current is the flow of electric charge in a circuit, measured in amperes (A). In this context, the current varies with time, indicating that it is not constant but changes according to the given equation. Understanding how current behaves over time is crucial for analyzing the total charge transported.

Charge (Q)

Charge is the quantity of electricity transported by the current, measured in coulombs (C). The total charge can be calculated by integrating the current over time. In this problem, we need to find a constant current that would transport the same amount of charge as the varying current over a specified time interval.

Integration

Integration is a mathematical process used to find the total accumulation of a quantity, such as charge over time. In this case, we will integrate the time-varying current function to determine the total charge transported during a specific time interval. This concept is essential for comparing the varying current to a constant current.