Worksheet 9: Helmholtz Coils and Earth's Magnetic Field¶

Introduction¶

In this lab, you will explore how magnetic fields interact with magnetic dipoles by using Helmholtz coils to generate a uniform magnetic field and observing the oscillatory motion of a magnet suspended in this field. The experiment provides a method to measure the magnetic dipole moment of a cylindrical magnet and the horizontal component of Earth’s magnetic field through simple harmonic motion (SHM) analysis.

Magnetic Torque and Simple Harmonic Motion¶

A magnetic dipole (e.g., a bar magnet) placed in a uniform magnetic field experiences a torque:

$\vec{\tau} = \vec{\mu} \times \vec{B}$

For small angular displacements $\theta$ , the torque becomes:

$\tau = \mu B \sin \theta \approx \mu B \theta$

This leads to the equation for angular SHM:

$\frac{d^2\theta}{dt^2} = -\frac{\mu B_{\text{net}}}{I} \theta$

where $I$ is the moment of inertia of the magnet and $B_{\text{net}} = B + B_E$ is the combined magnetic field from the Helmholtz coils and Earth's magnetic field. The frequency of oscillation is then:

$f^2 = \frac{\mu (B + B_E)}{4\pi^2 I}$

Since the magnetic field generated by the coils is proportional to the current $I$ , this gives a linear relationship:

$f^2 = mI + b$

This allows experimental determination of: - The magnetic dipole moment $\mu$ (from the slope), - The horizontal component of Earth’s magnetic field $B_E$ (from the y-intercept).

Objectives¶

By the end of this lab, you will be able to: - Construct and align a Helmholtz coil system to generate a uniform magnetic field. - Measure and analyze the oscillatory motion of a suspended magnet in a magnetic field. - Derive and use the equation for angular SHM involving magnetic torque. - Determine the magnetic dipole moment of a magnet from oscillation data. - Estimate the horizontal component of Earth’s magnetic field through experimental analysis.

Overview of Measurements¶

Measurement 1: Helmholtz Coil and Magnet Setup¶

You will measure the coil radius (outer and inner), the wire diameter, and calculate the average radius of the Helmholtz coils.
The mass and length of the magnet will be recorded to compute its moment of inertia.
The Helmholtz coil apparatus will be aligned with Earth’s magnetic field using a compass and adjusted so that both magnetic fields point in the same direction.
A range of currents (0.20 A to 1.20 A) will be applied to the coils.
For each current, you will:
Displace the suspended magnet by a small angle,
Time 20 complete oscillations,
Calculate the oscillation frequency and its square $f^2$ .

Analysis: Determining μ and Bₑ¶

You will plot $f^2$ vs. current $I$ and fit a straight line.
From the slope, you will calculate the magnet’s magnetic dipole moment $\mu$ .
From the y-intercept, you will determine the horizontal component of Earth’s magnetic field $B_E$ .
The analysis includes:
Calculating the moment of inertia for the cylindrical magnet,
Computing the field constant $C$ for the Helmholtz coils,
Discussing sources of uncertainty and alignment sensitivity.

Materials List¶

Helmholtz Coils
Cylindrical Magnet
Thread (to suspend the magnet)
DC Power Supply
Ruler, Digital Caliper, Digital Micrometer, Triple Beam Balance
Compass
Spreadsheet Software (Excel)