Estimating the EMF in the loop formed by your hand

Problem

Estimate the electromotive force induced in the loop formed by the outline of your hand (area $\sim 100 \ \mathrm{cm^2}$) if you close and open it at high speed, in the presence of (a) the Earth’s magnetic field ($B \sim 5 \cdot 10^{-5} \ \mathrm{T}$) and (b) a strong neodymium magnet ($B \sim 0{,}5 \ \mathrm{T}$) waved by a friend at $30 \ \mathrm{cm}$. Which of the two cases dominates?

Solution

Starting data (estimate). Loop area: $A \sim 100 \ \mathrm{cm^2} = 0{,}01 \ \mathrm{m^2}$; time to open/close the hand: $\Delta t \sim 0{,}1 \ \mathrm{s}$. The EMF follows from Faraday’s law: $\abs{\mathcal{E}} \sim \dfrac{\Delta \Phi}{\Delta t} \sim \dfrac{B \, A}{\Delta t}$.

(a) Earth’s field. The field is nearly constant and small; only the change in area contributes: $\abs{\mathcal{E}}_{\text{Earth}} \sim \frac{B \, A}{\Delta t} = \frac{5 \cdot 10^{-5} \cdot 0{,}01}{0{,}1} = 5 \cdot 10^{-6} \ \mathrm{V}$ Entirely negligible (a few microvolts).

(b) Neodymium magnet. Here it is not the change in area that matters but the change in field: waving the magnet changes $B$ through the hand by almost its full value, $\Delta B \sim 0{,}5 \ \mathrm{T}$, over a short time $\Delta t \sim 0{,}01 \ \mathrm{s}$ (rapid waving at close range): $\abs{\mathcal{E}}_{\text{magnet}} \sim \frac{\Delta B \, A}{\Delta t} = \frac{0{,}5 \cdot 0{,}01}{0{,}01} = \ev{0{,}5 \ \mathrm{V}}$

Conclusion. The magnet dominates by far: $0{,}5 \ \mathrm{V}$ against $5 \ \mathrm{\mu V}$, roughly four orders of magnitude more. The Earth’s field, although always present, is irrelevant compared with a strong magnet moved quickly near the loop.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Faraday-Neumann-Lenz law Competenze: Fermi estimates · Flux balance