Graphs of self-induction

Problem

An ideal solenoid carries a sinusoidal current $i(t) = I_0 \sin(\omega t)$ imposed by the generator. On the same plane (abscissa $t$, ordinates normalised to their respective peaks) draw: (a) the field $B(t)$ in the solenoid, (b) the self-induced EMF $\mathcal{E}_L(t) = -L\,\ddt{i}$, (c) the self-linked flux $\Phi_B(t)$. Indicate the phase shifts relative to $i(t)$.

Sinusoidal current imposed by the generator.

Solution

Field and flux. In the solenoid $B = \mu_0 \dfrac{N}{\ell} i \propto i$, so $B(t)$ is in phase with $i(t)$ (same sinusoid). The self-linked flux is $\Phi_B = L\,i \propto i$, so it too is in phase with $i(t)$: $\Phi_B(t) = L I_0 \sin(\omega t)$

Self-induced EMF. $\mathcal{E}_L(t) = -L\,\ddt{i} = -L I_0 \omega \cos(\omega t) = L I_0 \omega \,\sin\!\left(\omega t - \tfrac{\pi}{2}\right)$ The EMF is shifted by $\ev{+\tfrac{\pi}{2}}$ in lag (form $-\cos$): it is maximum in magnitude when $i$ crosses zero with maximum slope, and zero when $i$ is at its peaks.

Summary of phase shifts: $\ev{B \text{ and } \Phi_B \text{ in phase with } i; \quad \mathcal{E}_L \text{ shifted by a quarter period}}$.

In phase: $i$ and $\Phi_B$ (red). The EMF $\mathcal{E}_L$ (blue, $\propto -\cos$) is shifted by a quarter period.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Inductance and self-induction Competenze: Reading graphs Oggetti: Solenoid