The current in a wire varies with time according to the relationship I=55A−(0.65A/s2)t2I = 55 A - (0.65 $$A/s^2$)t^2I=55A$$−(0.65A/s2)t2 . What constant current would transport the same charge in the same time interval?

Ch 25: Current, Resistance, and EMF

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 25: Current, Resistance, and EMF

Problem 7b

Chapter 25, Problem 7b

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

Verified step by step guidance

First, understand that the problem involves finding a constant current that transports the same charge as a time-varying current over a given time interval.

To find the total charge transported by the time-varying current, integrate the current function I(t) = 55 A - (0.65 A/s^2)t^2 with respect to time over the interval from t = 0 to t = T, where T is the time interval.

The integral of the current function I(t) with respect to time will give the total charge Q transported: Q = ∫(55 - 0.65t^2) dt from 0 to T.

Calculate the integral: Q = [55t - (0.65/3)t^3] evaluated from 0 to T. This will give you the expression for the total charge transported by the time-varying current.

To find the constant current I_c that transports the same charge Q in the same time interval T, use the relationship Q = I_c * T. Solve for I_c by rearranging the equation: I_c = Q / T.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Current and Charge Relationship

Current (I) is the rate at which charge (Q) flows through a conductor. The relationship between current and charge is given by Q = ∫I dt, where the integral of current over time gives the total charge transported. Understanding this relationship is crucial for determining the equivalent constant current that transports the same charge over a given time interval.

Integration in Physics

Integration is a mathematical process used to find quantities like area under a curve, which in physics often represents accumulated quantities such as charge. In this context, integrating the current function I = 55 A - (0.65 A/s^2)t^2 over the specified time interval will yield the total charge transported, allowing comparison with a constant current scenario.

Constant Current

A constant current is one that does not change with time, unlike the given variable current. To find a constant current that transports the same charge, one must calculate the total charge transported by the variable current and then determine the constant current that would transport this charge over the same time interval, using the formula Q = I_constant × time.

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Textbook Question

The current in a wire varies with time according to the relationship $I = 55 A - (0.65 A / s^{2}) t^{2}$ . How many coulombs of charge pass a cross section of the wire in the time interval between $t equals 0$ and $t equals 8.0 s$ ?

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rank

The current in a wire varies with time according to the relationship I=55A−(0.65A/s2)t2I = 55 A - (0.65 A/s^2)t^2I=55A−(0.65A/s2)t2 . What constant current would transport the same charge in the same time interval?

Verified video answer for a similar problem:

Key Concepts

Current and Charge Relationship

Integration in Physics

Constant Current

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