Permanent Magnets

For a bar magnet with magnetic moment M in uniform field B at angle θ:

Torque: $\vec{\tau} = \vec{M} \times \vec{B}$ $\tau = MB\sin\theta$

This torque tries to align the magnet with the field (θ→0).

Potential Energy: $U(\theta) = -MB\cos\theta = -\vec{M}\cdot\vec{B}$

Key insights:

• Stable equilibrium: When magnet aligns with field (θ=0)
• Unstable equilibrium: When magnet anti-aligns with field (θ=180°)
• Maximum potential energy: When perpendicular to field (θ=90°)
• Work done rotating from θ₁ to θ₂: W = MB(cosθ₁-cosθ₂)

The SI unit for magnetic moment M can be expressed as J·T⁻¹.

Magnetic vs. Geometrical Length:

Magnetic length (2l): Distance between the locations of the assumed magnetic poles, which appear slightly inside the physical ends of the magnet.

Geometrical length: Actual physical length of the bar magnet from end to end.

Difference Explanation: The difference occurs because of end effects in the magnet. At the ends, the behavior of atomic currents differs from that inside the magnet, causing the effective poles to be slightly inset from the physical ends.

Typical Ratio: Magnetic length/Geometrical length ≈ 0.84

This ratio is important for accurate calculations of magnetic field strength and explains why magnetic poles appear to be located slightly inside the physical ends of any bar magnet.

The magnetic field at a point in the broadside-on position (perpendicular bisector) has:

• Magnitude: $B = \frac{\mu_0}{4\pi} \frac{m \cdot 2l}{(d^2 + l^2)^{3/2}} = \frac{\mu_0}{4\pi} \frac{M}{(d^2 + l^2)^{3/2}}$, where M = 2ml is the magnetic moment

• Direction: Parallel to the magnetic axis, pointing from the north pole to the south pole

• For a magnetic dipole approximation (when d >> l): $B = \frac{\mu_0}{4\pi} \frac{M}{d^3}$

This field results from the vector sum of the individual fields from both poles, with each contributing a component toward the axis of the magnet.

This technique corrects for the error caused when the 0°-0° line of the circular scale is not perfectly horizontal:

• When the dip needle is read in the initial position, the reading (δ₁) has an error due to the tilted reference line
• After rotating the dip circle 180° about the vertical axis, a second reading (δ₂) is taken
• The error in the first reading is equal in magnitude but opposite in direction to the error in the second reading
• Taking the average of these two readings: δ = (δ₁ + δ₂)/2 cancels out the error

Example: If true dip is 60° but the 0°-0° line is tilted 2° upward, the first reading might show 58° and the second 62°. The average (60°) gives the correct dip.

Note: This is part of a systematic process of error elimination in geomagnetic measurements.

The four main sources of instrumental errors in dip measurement are:

1. Misalignment of needle center with scale center

   • Corrected by reading both ends of the needle

2. Non-horizontal 0°-0° reference line

   • Corrected by rotating the circle 180° and taking new readings

3. Misalignment between magnetic and geometric axes of the needle

   • Corrected by inverting the needle on its bearing

4. Offset between center of mass and pivot point

   • Corrected by demagnetizing and remagnetizing the needle in opposite direction

A total of 16 readings are required for full error correction:

• 2 ends of needle × 2 positions of circle × 2 orientations of needle × 2 magnetization states

The final dip value is calculated as the average of all 16 readings.

A dip circle is an instrument used to measure the magnetic dip (inclination) at a location on Earth, consisting of:

• A vertical circular scale with 0° markings horizontally and 90° markings vertically
• A magnetic needle pivoted at the center that can rotate freely in the vertical plane
• A glass enclosure that can be rotated to align with the magnetic meridian

The instrument demonstrates how a freely suspended magnetic needle aligns itself with Earth's magnetic field lines in three dimensions, not just horizontally. When placed in the magnetic meridian, the needle's angle from horizontal directly indicates the dip angle.

Example application: Navigators historically used dip measurements to approximate their latitude when stellar observations weren't possible.

Moving-coil galvanometer:

• Operating principle: Current-carrying coil in permanent magnetic field experiences torque
• Deflection equation: $i = \frac{k}{nAB}θ$ (linear relationship)
• Sensitivity = $\frac{θ}{i} = \frac{nAB}{k}$
• Sensitivity improved by: Increasing magnetic field (B), number of turns (n), coil area (A), decreasing torsional constant (k)
• Advantages: Linear scale, higher sensitivity, measures small currents

Tangent galvanometer:

• Operating principle: Fixed coil produces field perpendicular to Earth's field, deflecting needle
• Deflection equation: $i = K \tan θ$ (non-linear relationship)
• Maximum sensitivity at θ = 45° where $\frac{dθ}{di}$ is maximum
• Sensitivity improved by: Decreasing coil radius (r), increasing turns (n)
• Disadvantages: Non-linear scale, affected by Earth's field variations, less sensitive

Key distinction: Moving-coil devices move the current-carrying element; tangent devices keep current element fixed and observe magnetic needle movement.

The deflecting torque in a moving-coil galvanometer arises from the Lorentz force on current-carrying conductors in a magnetic field:

• When current flows through the rectangular coil placed between magnetic poles, each vertical segment experiences a force perpendicular to both the current direction and magnetic field
• These forces form a couple, creating a torque
• Mathematically expressed as: Γ = niAB
  • n = number of turns in coil
  • i = current through coil
  • A = area of coil
  • B = magnetic field strength

This torque is balanced by the restoring torque of the suspension (k·θ), resulting in equilibrium at an angle θ where: niAB = k·θ, making i proportional to θ.

Experimental procedures for oscillation magnetometer measurements:

Initial setup:

1. Level the instrument using spirit level and leveling screws
2. Align the reference line with the magnetic meridian using a compass needle
3. Place a non-magnetic weight (brass) on the hanger to remove any twist
4. Adjust the torsion head to make the bar parallel to the reference line

Measurement procedure:

1. Gently place the magnet in the hanger with the north pole toward north
2. Set the magnet into angular oscillation with slight deflection
3. Count oscillations as the magnet crosses the reference line
4. Measure time for 20+ oscillations for greater accuracy
5. Calculate the time period T = total time/number of oscillations

Critical precautions:

• Keep the box closed during oscillations to prevent air currents
• Position your eye properly when observing (use mirror to avoid parallax)
• Keep the angular amplitude small to ensure simple harmonic motion
• Ensure no magnetic materials are nearby to avoid interference
• Measure the magnet's dimensions and mass accurately for correct I value
• Use an unspun silk thread to minimize torsional effects

Error reduction:

• Multiple readings should be taken and averaged
• Verify that oscillations maintain consistent amplitude
• Check for temperature stability throughout measurements

Center of Mass Error in Magnetic Dip Measurement

Problem:

• When the center of mass of a magnetic needle does not coincide with its pivot point
• The weight (mg) of the needle creates an additional torque
• This causes the needle to deviate from the true direction of Earth's magnetic field
• Result: Measured dip angle either exceeds or falls below the true dip angle, depending on center of mass position

Elimination Technique (Pole Reversal Method):

1. Take initial readings with the magnetic needle in its original state
2. Demagnetize the needle
3. Remagnetize the needle in the opposite direction (reversing N and S poles)
4. Take a second set of readings with the reversed polarity needle
5. Average all readings together

Why This Works:

• Reversing the poles keeps the center of mass in the same position
• The gravitational torque remains constant in direction
• The magnetic torque reverses direction
• The errors in the two sets of readings have opposite signs and cancel out when averaged

Note: This technique is typically part of a more comprehensive procedure involving 16 total readings to eliminate various instrumental errors.