the result.

Section $6\mathrm{.}2$

$6\mathrm{.}13$ The expressions for voltage, power, and energy derived in Example $6\mathrm{.}5$ involved both integration and manipulation of algebraic expressions. As an engineer, you cannot accept such results on faith alone. That is$,$ you should develop the habit of asking yourself, $"$Do these results make sense in terms of the known behavior of the circuit they purport to describe?" With these thoughts in mind, test the expressions of Example $6\mathrm{.}5$ by performing the following checks:

a$)$ Check the expressions to see whether the voltage is continuous in passing from one time interval to the next.

b$)$ Check the power expression in each interval by selecting a time within the interval and seeing whether it gives the same result as the corresponding product of $v$ and $i.$ For example, test at $10$ and $30s.$

c$)$ Check the energy expression within each interval by selecting a time within the interval and seeing whether the energy equation gives the same result as $\frac{1}{2}C{v}^2.$ Use $10$ and $30s$ as test points.

$6\mathrm{.}15$ The current and voltage for a $0\mathrm{.}6F$ capacitor are both zero for For $t0,$ the current is $3cos50,000tA.$

a$)$ Find the expression for the voltage drop across the capacitor in the direction of the current.

b$)$ Find the maximum power delivered to the capacitor any one instant in time.

c$)$ Find the maximum energy stored in the capacitor any one instant in time.