Physics

Problem

A precision function generator drives a current

I(t)

through a coil according to the sinusoidal relation

I(t) = I_{\text{peak}} \sin(\omega t)

, starting at

t=0

. • The peak current

I_{\text{peak}}

is 2.5 A. • The angular frequency

\omega

120\pi\,\text{rad}\,\text{s}^{-1}

(corresponding to a 60 Hz power signal). (a) Derive an algebraic expression for the total charge

q(t)

that has passed through the coil from time

t=0

up to an arbitrary time

t

. (b) Using your result from part (a), calculate the exact amount of charge that passes through the coil during the first quarter of a complete cycle. (c) Determine the earliest time

t_{1\,\text{C}}

at which exactly

1.0\,\text{C}

of charge has passed through the coil. (d) Over a full cycle (period

T

), what is the net charge transported through the coil? Explain physically why this result makes sense in the context of alternating current.

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