Question

The charge entering the positive terminal of an element is q = 8 sin(4ππt) mC, while the voltage across the element (plus to minus) is v = 3 cos(4ππt) V.

Required information. Find the power delivered to the element at t = 0.3 s. The power delivered to the element at t = 0.3 s is mW.

Answer 1

To find the power delivered to the element at t = 0.3 s, differentiate the charge function to get the current, then substitute t = 0.3 s into both the current and voltage functions and use P = IV.

Step-by-step explanation:

To calculate the power delivered to a circuit element at a given time, you need both the voltage across the element and the current through it. The power equation, P = IV, requires instantaneous values of current (I) and voltage (V). The provided voltage function is v(t) = 3 cos(4πt) V, and the charge function is q(t) = 8 sin(4πt) mC. To obtain the current, differentiate the charge with respect to time: i(t) = dq(t)/dt. Doing this for the given charge function yields i(t) = d/dt [8 sin(4πt)] mC = 32π cos(4πt) mA because the derivative of sine is cosine and the factor 4π comes out due to the chain rule. At t = 0.3 s, this becomes i(0.3) = 32π cos(4π× 0.3).

Now, substitute t = 0.3 s into both i(t) and v(t) to find their instantaneous values. After that, use the power formula P = IV with these values to find the power delivered to the element at t = 0.3 s. The result will be in milliwatts because the current was calculated in milliamps.