Helmholtz coils are two identical circular coils having the same radius 𝑅 and the same number of turns N, separated by a distance equal to the radius 𝑅 and carrying the same dc current I in the same direction. (See Fig. 28–61.) They are used in scientific instruments to generate nearly uniform magnetic fields. (They can be seen in the photo, Fig. 27–19.) (a) Determine the magnetic field B at points 𝓍 along the line joining their centers. Let 𝓍 = 0 at the center of one coil, and 𝓍 = 𝑅 at the center of the other. (b) Show that the field midway between the coils is particularly uniform by showing that dB/d𝓍 = 0 and d²B/d𝓍² = 0 at the midpoint between the coils. (c) If 𝑅 = 10.0 cm, N = 85 turns and I = 3.0 A, what is the field at the midpoint between the coils, 𝓍 = 𝑅/2 ?


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Verified step by step guidance

1. Identify the formula for the magnetic field at the center of a single circular coil, which is given by B = $\frac{{\mu }_{0}NI}{2R}$ (where ${\mu }_0$ is the permeability of free space, N is the number of turns, I is the current, and R is the radius of the coil).
2. Apply the superposition principle to calculate the total magnetic field at any point along the axis, considering the contributions from both coils. Since the coils are identical and carry the same current, the magnetic fields due to each coil will add up.
3. To find the magnetic field at the midpoint (x = R/2), use symmetry and the fact that the contributions from each coil will be equal and in the same direction at this point.
4. To show that the field is particularly uniform at the midpoint, calculate the first derivative (dB/dx) and the second derivative (d²B/dx²) of the magnetic field with respect to x at x = R/2. Show that both derivatives are zero, indicating a maximum or minimum point, which in this context is a point of uniformity.
5. For the specific values given (R = 10.0 cm, N = 85 turns, I = 3.0 A), substitute these into the formula derived for the magnetic field at the midpoint to find the magnetic field strength there.

Key Concepts

Magnetic Field of a Coil

The magnetic field generated by a circular coil of wire is determined by the current flowing through it and the number of turns in the coil. The magnetic field at the center of a single loop is given by the formula B = (μ₀ * I) / (2 * R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the coil. For Helmholtz coils, the configuration allows for the superposition of the magnetic fields from both coils, leading to a more uniform field in the region between them.

Superposition Principle

The superposition principle states that when multiple magnetic fields are present, the total magnetic field at any point is the vector sum of the individual fields. In the case of Helmholtz coils, the magnetic fields produced by each coil add together along the axis between them. This principle is crucial for analyzing the resultant magnetic field and understanding how it varies with position along the line connecting the coils.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength (B) is constant in magnitude and direction throughout a specified region. For Helmholtz coils, the design aims to create a uniform field in the space between the coils, particularly at the midpoint. To demonstrate uniformity mathematically, one can show that the first derivative of the magnetic field with respect to position (dB/dx) is zero, indicating no change in field strength, and the second derivative (d²B/dx²) is also zero, confirming that the field does not curve or vary in that region.