The current in a wire varies with time according to the...

The current in a wire varies with time according to the relationship $I=55 \mathrm{~A}-\left(0.65 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}$. (a) How many coulombs of charge pass a cross section of the wire in the time interval between $t=0$ and $t=8.2 \mathrm{~s} ?$ (b) What constant current would transport the same charge in the same time interval?

Key Concepts

Relationship between Current and Charge

Current is defined as the rate of flow of electric charge with respect to time. This means that the charge that passes through a cross-section of a conductor over a given time interval can be found by integrating the current function with respect to time.

Definite Integration

Definite integration is a central concept in calculus that calculates the accumulation of a quantity, such as charge, over a specified interval. In the context of this problem, integrating the current function from the initial to the final time yields the total charge transferred.

Average Value of a Function

The average value of a function over an interval is determined by dividing the total accumulated quantity by the duration of the interval. In this scenario, calculating the average current that transports an equivalent amount of charge as a variable current involves dividing the total charge by the time interval, thereby finding a constant current that yields the same total charge transfer.

Transcript

00:01 The current in a wire varies with time according to the relation i is equal to here i is equal to 55 ampiers minus 0 .65 ampers per second square multiplied by t square so this is a relation right how many coulamps of charge pass a cross section of while in time interval between 3 equals to 0 to 8 .2 seconds.

00:46 Right.

00:47 Is that the same one? right.

00:49 So now this is 8 .2 seconds.

00:55 So yes.

00:56 So now here, how many are like, okay.

00:59 So this is first question it is asked.

01:02 Secondly it is asked, what constant current would transport the same charge in the same time interval? right so now here this is given to us let us just suppose this as i is the function of t here i is function of t right so now since charge is basically given by q equals to integration of i t d t right and the limits are from zero to to t here right and the limits are from zero to t here right now, here t is equal to 8 .2 seconds.

01:51 So it makes it 0 to 8 .2 seconds.

01:57 Right.

01:57 So it is 55 minus 0 .65 t squared.

02:06 And this will multiply by d t.

02:09 So this is giving us q is equal to 55.

02:16 Right it should be t multiplied by t so upper limit minus lower limit that means 8 .2 minus 0 0 .65 divided by 3 because t cube by 3 it becomes right so this is multiplied by t cube so this will be 8 .2 cube minus 0 cube right so in all total it makes said q is equal to let us just see it is just calculate for it it becomes 55 multiplied by 8 .2 which is giving us the value 451 minus 451 minus 8 .2 cube right and this is whole multiplied by 0 .655 right and this is the whole divided by 3.

03:35 So this is making it 119 .463 almost right.

03:42 This is giving us the value q is equal as to 331 .5...

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