Problem 67 - University Physics with Modern Physics

Question

Helmholtz Coils. Fig. 28.59 is a sectional view of two circular coils with radius \(a\) , each wound with \(N\) turns of wire carrying a current \(I,\) circulating in the same direction in both coils. The coils are separated by a distance \(a\) equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude \(B\) of the magnetic field at a point on the axis a distance \(x\) to the right of point \(P,\) which is midway between the coils. (b) Graph \(B\) versus \(x\) for \(x=0\) to \(x=a / 2\) . Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point \(P .\) (d) Calculate the magnitude of the magnetic field at \(P\) if \(N=300\) turns, \(I=6.00 \mathrm{A},\) and \(a=8.00 \mathrm{cm} .\) ( e) Calculate \(d B / d x\) and \(d^{2} B / d x^{2}\) at \(P(x=0)\) . Discuss how your results show that the field is very uniform in the vicinity of \(P .\)

Step-by-Step Solution

Answer

The magnetic field at point P is more uniform with Helmholtz coils, and at P, it evaluates to a specific constant value based on given parameters.

Step 1: Calculate the Magnetic Field from a Single Coil

For a single circular loop carrying a current, the magnetic field at a distance \( x \) along the axis from the center of the coil is given by \[ B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \]where \( R \) is the radius of the loop, \( I \) is the current, and \( \mu_0 \) is the permeability of free space. For \( N \) coils, the field becomes \[ B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}. \]

Step 2: Derive the Expression for Helmholtz Coils

Now, we consider the contributions from both coils. They are separated by a distance \( a \) (equal to the radius \( a \)). The magnetic field at a point \( x \) from the midpoint due to both coils is the sum of the fields from each coil. - For the left coil, the point is at distance \( (a/2 + x) \) from the coil.- For the right coil, the point is at \( (a/2 - x) \).Combining these, \[ B = \frac{\mu_0 N I a^2}{2[(a^2 + (a/2 + x)^2)^{3/2} + (a^2 + (a/2 - x)^2)^{3/2}]} \].

Step 3: Graph the Magnetic Field versus Distance

Plot \( B \) as a function of \( x \) from \( 0 \) to \( a/2 \) using the above expression. Comparatively, plot the magnetic field from the right coil alone given by \[ B = \frac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}} \]. The graph should show that the field from Helmholtz coils is more uniform than from a single coil.

Step 4: Calculate the Magnetic Field at Point P

Point \( P \) is at \( x = 0 \), so use the derived formula \[ B_P = 2 \times \frac{\mu_0 N I a^2}{2(a^2 + (a/2)^2)^{3/2}} = \frac{\mu_0 N I a^2}{(a^2 + a^2/4)^{3/2}} \] and simplify.

Step 5: Compute the Value of the Magnetic Field at P with Given Values

Substitute \( N = 300 \), \( I = 6.00 \mathrm{A} \), and \( a = 0.08 \mathrm{m} \) into the above equation and calculate the magnetic field \( B_P \).

Step 6: Calculate Derivatives \( dB/dx \) and \( d^2B/dx^2 \) at P

Evaluate the first and second derivatives of \( B \) at \( x = 0 \). - Compute \( dB/dx \) by differentiating the derived expression with respect to \( x \).- Compute \( d^2B/dx^2 \) similarly.Assess the uniformity by noting that \( dB/dx = 0 \) and \( d^2B/dx^2 \) is small at \( P \).

Key Concepts

Magnetic Field Uniformity

Helmholtz coils are specifically designed to create a region where the magnetic field is extremely uniform. Uniform magnetic fields are important in experiments requiring precise magnetic environments, such as those using atomic or magnetic resonance. This uniformity is achieved because the two coils are symmetrically placed about the point of interest and are at a distance from each other equal to their radius.
To analyze the uniformity, we look at the rate of change of the magnetic field as you move away from the center point. When you calculate the first derivative \(dB/dx\) of the magnetic field with respect to the distance from the midpoint, you will find that at the center (where \(x = 0\)), this derivative equals zero. This implies that the slope of the magnetic field is flat at this point, contributing to uniformity.

• Helmholtz configuration: Two coils, same current direction, separated by one radius.
• Main advantage: Very uniform field in the center area.
• Analytical proof: Derivative \(dB/dx = 0\) at midpoint indicates field stability.

Moreover, a small second derivative \(d^2B/dx^2\) suggests that the change in the field is gradual around the center, reinforcing the concept of uniformity under small displacements. Hence, Helmholtz coils are excellent for applications needing stable and predictable magnetic fields.

Magnetic Field Calculation

Calculating the magnetic field produced by Helmholtz coils at a point involves a clear understanding of electromagnetic principles. At the core, magnetic fields are produced by moving charges, and in the case of Helmholtz coils, they are due to the current flowing through loops of wire.
The fundamental formula used to calculate the magnetic field from a single loop is:\[ B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}} \]where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( I \) is the current, and \( x \) is the distance along the axis.To find the field at a point due to two Helmholtz coils, we add the fields due to each coil:

• Field from left coil: Point is at \((a/2 + x)\) from the midpoint.
• Field from right coil: Point is at \((a/2 - x)\) from the midpoint.

Resulting in the total magnetic field being given by:\[ B = \frac{\mu_0 N I a^2}{2[(a^2 + (a/2 + x)^2)^{3/2} + (a^2 + (a/2 - x)^2)^{3/2}]} \]Calculating these expressions can seem complex, but they essentially tell us how the field balances out due to symmetry and geometry. When properly arranged, as in Helmholtz coils, both the direction and magnitude interplay to produce a stable magnetic field.

Electromagnetism

Electromagnetism is a fundamental principle underlying the operation of Helmholtz coils. In essence, it studies how electric currents interact with magnetic fields and how charges react to magnetic forces. At its core, electromagnetism is one of the four fundamental forces of nature and is crucial for technologies ranging from motors to data storage.
Helmholtz coils capitalize on the laws of electromagnetism to produce a uniform magnetic field. The coils work based on Ampère’s Law, which states that electric currents create magnetic fields. Further, the Biot-Savart Law provides the quantitative means to calculate these fields in precise conditions such as a coil setup.

• Ampère's Law: Connects electricity to magnetism by showing how currents create fields.
• Biot-Savart Law: Helps calculate field strength at various points in space.
• Application: Ensures precise creation and manipulation of fields for experiments.

The arrangement of the Helmholtz coils ensures that the various field components merge to create a highly predictable and uniform field pattern. This perfect balance makes electromagnetism at the helm of controlling environments where predictable magnetic influence is crucial, demonstrating once more the power and application of this natural force.