Question

Now consider an inductor with inductance l in an ac circuit (diagram 2). Assuming that the current in the inductor varies as il(t)=i0cos(Ωt), find the voltage vl(t) that must be driving the inductor?

Answer 1

To determine the voltage across an inductor in an AC circuit with a current of i0cos(Ωt), use the differentiated current to find VL(t) = Li0Ωcos(Ωt + π/2), considering the 90° phase lead of voltage over current.

Step-by-step explanation:

The question relates to finding the voltage vL(t) across an inductor in an AC circuit when the current iL(t) varies as i0cos(Ωt). To find this voltage across the inductor, we can use the formula VL(t) = -L(dI/dt). Given the current expression iL(t) = i0cos(Ωt), we differentiate it with respect to time to obtain dI/dt and then calculate vL(t). If we perform this differentiation, we have dI/dt = -i0Ωsin(Ωt). Substituting this back into the formula for VL(t), we get VL(t) = L(i0Ωsin(Ωt)) = Li0Ωsin(Ωt). Because the voltage leads the current by 90° in an AC circuit with an inductor, the final expression for the voltage across the inductor driven by the AC source is vL(t) = Li0Ωcos(Ωt + π/2).