Unravelling the Mysteries of the Rogowski Coil

[togahockey15]

Homework Statement

When a wire carries an AC current with a known frequency you can use a Rogowski
coil to determine the amplitude Imax of the current without disconnecting the wire to
shunt the current in a meter. The Rogowski coil, shown in the figure, simply clips
around the wire. It consists of a toroidal conductor wrapped around a circular return
cord. The toroid has n turns per unit length and a cross-sectional area A. The current
to be measured is given by I(t) = Imax sin (ω t). (a) Show that the amplitude, E, of the
emf induced in the Rogowski coil is E = μ0 n A ω Imax. (b) Explain why the wire
carrying the unknown current need not be at the center of the Rogowski coil, and why
the coil will not respond to nearby currents that it does not enclose.

*The figure is simply a ring of wire that has another wire wrapped around it, with a current going through the ring.

Homework Equations

Emf= -N(dI/dt) where I = magnetic flux, not current

Emf = I/R = R*(dQ/dt) where I = current

Magnetic flux = *integral* (B*dA)

Emf = *surface integral* (E*dL) = -(d*magnetic flux*/dt)

The Attempt at a Solution

I am not really sure where to start - maybe by using the Emf= I/R ?? Can anyone help get me started in the right direction? Thanks!

 

[Jhero]

Thank you for your interesting post. I am happy to help you understand how to determine the amplitude of an AC current using a Rogowski coil.

To begin, let's review the equations you have listed. The first equation, Emf= -N(dI/dt), is correct. This is known as Faraday's Law, which states that the induced electromotive force (emf) in a closed loop is equal to the negative of the time rate of change of magnetic flux through the loop.

The second equation, Emf = I/R = R*(dQ/dt), is also correct. This is Ohm's Law, which relates the emf to the current and resistance in a circuit.

Next, let's look at the magnetic flux equation, which is *integral* (B*dA). This equation represents the magnetic flux through a surface, which is the sum of all the magnetic field lines passing through that surface. The units of magnetic flux are webers (Wb).

Finally, we have the equation Emf = *surface integral* (E*dL) = -(d*magnetic flux*/dt). This equation is known as the Maxwell-Faraday equation, which relates the emf to the electric field and the change in magnetic flux over time.

Now, let's apply these equations to the Rogowski coil. The Rogowski coil is essentially a loop of wire with a circular return cord. When a current is passed through the wire, it creates a magnetic field around the wire. This magnetic field will pass through the circular return cord, creating a changing magnetic flux over time.

Using Faraday's Law, we can calculate the induced emf in the Rogowski coil by taking the negative of the time rate of change of the magnetic flux. This emf will be proportional to the number of turns in the coil, the cross-sectional area of the coil, and the frequency of the AC current. Therefore, we can write the equation as E = -N(dΦ/dt), where N is the number of turns per unit length, Φ is the magnetic flux, and t is time.

Next, we can use Ohm's Law to relate the emf to the current and resistance in the coil. The resistance of the coil is negligible, so we can write the equation as E = I/R. Therefore, we can rewrite the equation as E = (N/R)(dΦ/dt