Question

The current flowing through a 10 micro Farad capacitor is given by I (t) 0 across capacitor 0.3 exp(-2,000 t) A for t is greater or equal to 0 s. Given that the voltage t = 0 across the capacitor is zero, find the expression for v (t).

a) V (t) 10⁵ * exp (-2000t) V
b) V(t) 10⁵ * exp (-2000t)+ 0.3 V
c) V (t) 10⁵ * exp (-2000t)-0.3 v
d) V (t) = 10⁵ * exp (-2000t) V
e) None of the above

Answer 1

To find the voltage v(t) across a capacitor, we integrate the current I(t) with respect to time. The given current function I(t) 0.3 exp(-2000 t) A, when integrated and considering the initial condition v(0) = 0, leads to a voltage expression that does not match any of the provided options, making the correct answer 'None of the above'.

Step-by-step explanation:

To find the expression for the voltage v(t) across the capacitor, we need to integrate the current I(t) with respect to time. Given that the initial voltage at t=0 is zero and the current I(t) is 0.3 exp(-2000 t) A, the voltage across the capacitor v(t) can be found by integrating the current over time.

The formula for the voltage across a capacitor is given by v(t) = \(\frac{1}{C}\int I(t) dt\), where C is the capacitance. Substituting the given values into this equation, we get:
v(t) = \(\frac{1}{10 \times 10^{-6} F}\int 0.3 e^{-2000 t} dt\)

Integrating the current with respect to time gives us:
v(t) = 30 \times 10^{6} \times \(-\frac{1}{2000}\) e^{-2000 t} + K

Since we know that v(0) = 0, we can solve for the constant K. Doing so, we find that K = 0, and therefore the voltage v(t) is:

v(t) = 30 \times 10^{6} \times \(-\frac{1}{2000}\) e^{-2000 t}

Simplifying, we obtain the expression for v(t) as:

v(t) = -15 e^{-2000 t}

However, since voltage cannot be negative in this context, we take absolute value and find that:

v(t) = 15 e^{-2000 t}

Comparing this result with the given options, it turns out none of the listed options match the correct expression. Thus, the correct answer is: