The lying voltmeter

Problem

A voltmeter is connected to two points $A$ and $B$ of a loop carrying a varying magnetic flux. Changing the routing of the leads around the loop changes the reading. Explain in five lines why this apparent absurdity is a direct consequence of Faraday’s law.

Solution

The electric field is no longer conservative. With a varying magnetic flux, Faraday’s law in integral form states: $\oint \vec E \cdot \dd \vec\ell = -\frac{\dd \Phi}{\dd t} \neq 0$ The circulation of the electric field along a closed loop is not zero. This means $\vec E$ does not derive from a potential: there is no unique p.d. between $A$ and $B$.

Why the lead routing matters. The voltmeter reading is the integral of $\vec E$ along the actual path of the leads. Two different paths connecting $A$ to $B$ enclose different areas, and hence link different fluxes. Their difference is exactly $\dd \Phi / \dd t$ of the area enclosed between the two paths: $V_1 - V_2 = -\frac{\dd \Phi_{\text{enclosed}}}{\dd t}$

Conclusion. This is not a malfunction: in the presence of a varying flux, the “voltage between two points” loses its absolute meaning and becomes path-dependent. That is exactly what Faraday’s law predicts.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Faraday-Neumann-Lenz law · Magnetic flux