# How to measure voltage and current in closed circuit under magnetic induction?

Let's have a long closed loop wire, part of which is a coil to increase locally the inductance. A magnet is moved back and forth inside the coil.

A clamp amperimeter far from the coil, in the AC mode, shows that there is a current in the circuit.

My first doubt is: What is really detected is a magnetic force, and the system interprets it as produced by a current. According to the RHS of the Ampere-Maxwell law, the magnetic field can also result from the changing electric field. How the device knows the contribution of each term of the RHS?

\begin{align} \oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \left(\iint_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} + \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} \right) \\ \end{align}

And that electric field is indeed produced according to the Maxwell-Faraday law:

$$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = - \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S}$$

What leads me to another doubt. How to measure $$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}$$

A AC voltimeter will show a voltage between 2 points if the circuit is open. But it changes the assumption of a closed circuit and closed integral. And between 2 points of the closed conductor the voltage is zero.

How the device knows the contribution of each term of the RHS?

It doesn't, it does not work like that. The usual AC clamp ampermeter is based on current transformer. The measured conductor is a "primary winding" with about one coil and the measuring device contains the secondary winding in the clamp. When AC current is present in the measured conductor (the primary), it induces EMF in the secondary winding and that produces current that is measured inside the meter. So the measurement is based on the understanding of the effect of oscillating current in the primary on the induced electric current in the secondary, which is based on Faraday's law and details of the transformer in the meter.

Magnetic field is measured directly only in DC clamp meters using Hall probe. There is no displacement current then, magnetic field is determined by current only.

How to measure $$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}$$

A AC voltimeter will show a voltage between 2 points if the circuit is open. But it changes the assumption of a closed circuit and closed integral. And between 2 points of the closed conductor the voltage is zero.

Yes, introducing voltmeter into the circuit will change the circuit - introduce large impedance into it, so the measured system is changed. If we just take a voltmeter and standard non-shielded cables, connect the probes to measured points, then the voltmeter is really measuring the loop integral above, but for the system just created, that includes the voltmeter. One can demonstrate effect of induced EMF on the voltmeter reading this way.

If electric field is changing slowly enough, the voltmeter then shows some value that the total EMF (including electrostatic and induced electric field) impacts on the voltmeter inside. This in general is not a very useful measurement because it depends on positioning and shielding quality of the probe cables (their mutual inductance with the source of changing magnetic field) and other details such as construction of the voltmeter.

The closed loop integral of total electric field for a circuit without voltmeter or ampermeter is really a theoretical concept - total induced EMF in a circuit. Measuring it directly by measuring electric field at all points of the circuit is in principle possible, but it will be hard to do and I don't know of any case where it is routinely done. Most meaningful measurements of electric systems using voltmeters/oscilloscopes really are about measuring voltage between two points (difference of potential), not total EMF through some closed loop (because that is arbitrary). Measuring the correct voltage instead of random EMF is achieved by making a circuit with the voltmeter/oscilloscope using shielded coaxial probe cables with minimum loops so any induced field effect (EMF) on the measurement is neutralized.