Measuring magnetic fields

When a coil of wire is inserted into the aperture of a magnet and subjected to a changing magnetic environment, an induced voltage appears across its terminals. This process is governed by Faraday’s Law of Electromagnetic Induction, which states that the electromotive force (emf) or voltage induced in a coil is proportional to the rate of change of the magnetic flux through the coil. For a coil with $N$ turns, the induced emf is:

$$\text{emf}=-N\frac{d\Phi }{dt}$$

Here, $\Phi$ represents the magnetic flux, a quantity that measures the total magnetic field ($B$) passing through the area ($A$) of the coil. Flux is calculated as:

$$\Phi =\int B\cdot dA$$

The negative sign reflects Lenz’s law: the induced voltage always acts to oppose the change in flux. In practice, moving the coil through a magnet aperture causes a change in magnetic flux linkage, inducing a measurable voltage. By integrating this voltage over the time interval during which the coil experiences changing flux, one obtains:

$$\int \text{emf}dt=-N\Delta \Phi$$

This gives the total change in magnetic flux linkage for the coil.

Relating flux to magnetic field, if the magnetic field inside the aperture is uniform and the coil area is well known, the average field experienced by the coil is simply the total flux divided by the coil’s area:

$$B_{\text{avg}}=\frac{\Phi }{A}$$

This value characterizes the mean field encountered by the coil across its entire area as it passes through or resides within the magnet.

In contrast, the integral field (or field integral) is the sum of the magnetic field along the path of the coil through the aperture:

$$\int Bdl$$

where $dl$ is a differential element along the coil’s path. In accelerator physics, this field integral is directly related to the deflection experienced by a charged particle traversing the magnet: the particle’s bending angle depends on the field integral, not just the average field alone. For a uniform field over a path length $L$, the field integral reduces to $B\cdot L$, but for variable fields, the integral accounts for all variations.

Integral field measurement

When measuring the integral field of a magnet with a coil, the relevant length $L$ in expressions such as $B\cdot L$ is determined by the region over which the coil interacts with the magnetic field, rather than the mechanical length of the magnet itself. The measurement collects contributions not only from the central region inside the magnet but also from the fringe fields present at the magnet’s entrance and exit. If a measurement coil is longer than the magnet, it will begin to intercept the magnetic field before fully entering the main pole region and continue to detect field even after departing from it. The total measured field integral therefore becomes ${\int }_{l_1}^{l_2}B(l)dl$ where the limits $l_1$ and $l_2$ are set by the spatial extent of the coil as it travels through the magnet and its fringes. The result incorporates both the uniform central field and the weaker fringe fields at the ends. The effective length often referred to in such measurements is defined so that $B_{\text{eff}}\cdot L_{\text{eff}}={\int }_{l_1}^{l_2}B(l)dl$ where $B_{\text{eff}}$ is the average field over the region actually sampled by the coil, and $L_{\text{eff}}$ is the corresponding length, which may be larger than the physical length of the magnet when fringe effects are included. The geometry and length of the coil directly influence which portions of the field profile are included in the measurement. This approach ensures that all magnetic contributions relevant to a charged particle or probe passing through the magnet aperture, including those from the fringe regions, are correctly accounted for in the integral field measurement.

Distinguishing Use Cases in Accelerator Physics

The average field tells how strong the field is, on average, across a specified region, making it useful for characterizing the magnet’s uniformity and for calibration over known areas. The integral field, on the other hand, is crucial for understanding the overall effect a magnet will have on a charged particle’s trajectory, especially when the particle travels through non-uniform fields or along complex paths. In practice, the field integral determines key quantities like the momentum imparted to a particle beam and the total deflection angle.

For example, if a particle traverses a dipole magnet with a uniform field, both the average field and the integral field (using the magnet’s effective length) are relevant and closely related. If the field varies along the path, or in multipole magnets where the field’s spatial dependence is critical, the field integral captures these contributions. Therefore, in beam steering and focusing applications, the field integral is often the primary quantity of interest, while the average field may be more useful for evaluating magnet performance, uniformity, or quality control.

In summary, measuring the time-integrated induced voltage in a search coil provides access to the total magnetic flux (and thus the average field if area is known). By relating this flux change to the path of the coil, one can also evaluate the field integral, a quantity of primary importance for determining particle trajectories and the overall effectiveness of magnetic elements in accelerator physics.