Find the power delivered to the element at t = 5 ms if the current remains the same but the voltage is: (a) v = 2i V, (b) v = (10 + 5 ∫(0 to t) i dt) V.

Steps to Solve

1. Understanding Power Calculation To find power ($P$), we use the formula: $$ P = V \cdot I $$ where $V$ is the voltage and $I$ is the current.

2. Finding the Power for (a) For part (a), the voltage is given as: $$ V = 2i $$ At $t = 5 \ \text{ms}$, we calculate the power: $$ P_a = V \cdot I = (2i) \cdot I = 2I^2 $$

3. Determine Current (I) at t = 5 ms We need the value of the current $I$. Assuming it is provided as 5 A, we substitute: $$ P_a = 2(5)^2 = 2 \cdot 25 = 50 \ \text{W} $$

4. Finding the Power for (b) For part (b), the voltage is given as: $$ V = 10 + 5 \int_{0}^{t} i , dt $$ Substituting $t = 5 \ \text{ms}$ (which is 0.005 s), we will need the integral of the current over this time.

5. Calculate the Integral Assuming we know the current function, let’s say $i(t) = 5 \ \text{A}$ (constant current): $$ \int_{0}^{0.005} i , dt = i \cdot t = 5 \cdot 0.005 = 0.025 \ \text{A s} $$

6. Substituting the Integral Value Substituting back into the voltage equation: $$ V = 10 + 5 \cdot 0.025 = 10 + 0.125 = 10.125 \ \text{V} $$

7. Calculating Power for (b) Now calculate the power: $$ P_b = V \cdot I = 10.125 \cdot 5 = 50.625 \ \text{W} $$