What is the total energy?

Question

The current and voltage across a circuit element are given, respectively, by the equations  I (t) = 20e^-100t , V (t) = 2-e^-100t . 

Determine the total energy dissipated by the element between t = 0 and t = 10 ms.

Answer 1

Assuming the functions have "t" represented as seconds:

E = ₀∫^0.01 (20e^-100t)(2 - e^-100t) dt

E = ₀∫^0.01 (40e^-100t - 20e^-200t) dt

E = [(40/-100)e^-100t - (20/-200)e^-200t] evaluated between 0 and 0.01

E = [-0.4e^-100t + 0.1e^-200t] evaluated between 0 and 0.01

E = [-0.4e^-100(0.01) + 0.1e^-200(0.01)] - [-0.4e^-100(0) + 0.1e^-200(0)]

E = [-0.4e^-1 + 0.1e^-2] - [-0.4 + 0.1]

E = [-0.4/e + 0.1/e²] - [-0.3]

E = -0.14715 + 0.01353 + .3

E = 0.166 joules