The charge entering the positive terminal


The charge entering the positive terminal of an element is q(t)=-30e^{-4t} mC. If the voltage across the element is 120e^{-2t} V, determine the energy delivered to the element in the time interval 0 < t < 50 ms.

Solution:

We can using the following formula to figure out the energy delivered to the element:

\,\displaystyle W=\int (v)(i) dt

(Where W is energy, v is voltage and i is current)

 

To use the formula, we need to figure out the equation for i. Remember that:

i=\dfrac{d\,(q(t))}{dt}

(Substitute our q(t) equation)

i=\dfrac{d(-30e^{-4t})}{dt}

(Take the derivative)

i=120e^{-4t} mA

(Remember that our initial q(t) equation had units of mC. Thus, the current equation gives us mA. To get A, we need to divide our equation by 1000)

i=0.12e^{-4t} A

 

We can now substitute this equation into our previous integral along with the voltage equation.

\,\displaystyle W=\int (v)(i) dt

\,\displaystyle W=\int^{0.05}_{0} (120e^{-2t})(0.12e^{-4t}) dt

(Remember, 50 ms = 0.05 s)

W=-2.4e^{-6t}\Big|^{0.05}_{0}

W=0.622\,J

 

This question can be found in Basic Engineering Circuit Analysis, 10th edition, chapter 1, question 1.10.

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