Permanent Magnets

When a compass needle is subjected to two perpendicular magnetic fields:

1. The needle aligns itself with the resultant magnetic field vector (Br)
2. The resultant field is the vector sum of:
   • The horizontal component of Earth's magnetic field (BH)
   • The external perpendicular magnetic field (B)

The final position is determined by:

• The needle experiences forces mBr on its north pole and -mBr on its south pole
• These forces produce zero net torque only when the needle aligns parallel to Br
• The equilibrium angle θ between the needle and the original direction is given by: $\tan\theta = \frac{B}{B_H}$

This principle is utilized in tangent galvanometers and other magnetic measuring instruments.

Using the tangent law of perpendicular magnetic fields:

Step 1: Identify the formula relating the fields and deflection angle: $B = B_H \tan\theta$

Step 2: Substitute the known values:

• BH = 3.0 × 10^-5 T
• θ = 30°

Step 3: Calculate the unknown field: $B = (3.0 × 10^{-5} \text{ T}) \times \tan(30°)$ $B = (3.0 × 10^{-5} \text{ T}) \times (0.577)$ $B = 1.73 × 10^{-5} \text{ T}$

The perpendicular magnetic field strength is 1.73 × 10^-5 T.

Note: This principle is fundamental to field measurement techniques and explains why a compass needle deviates in the presence of external magnetic sources.

A soft-iron core is included in a moving-coil galvanometer to:

1. Concentrate and intensify the magnetic field in the gap between the pole pieces
2. Provide a low-reluctance path for magnetic flux, increasing field strength without requiring a larger permanent magnet
3. Create a more uniform radial field distribution around the coil

This design significantly enhances sensitivity because:

• The sensitivity equation is: sensitivity = nAB/k
• Increasing B directly improves sensitivity
• The stronger field produces greater torque for a given current
• Lower currents produce measurable deflections, making the instrument capable of detecting very small currents

The soft-iron core thus transforms what would be a moderately sensitive instrument into a high-precision measuring device suitable for detecting microampere-level currents.

The Tan-A position in a deflection magnetometer is characterized by: • The arms of the magnetometer are aligned along the magnetic east-west direction • The bar magnet is placed parallel to the arm's length (east-west orientation) • The compass needle is in "end-on" position relative to the bar magnet • The distance 'd' is measured from the center of the compass to the center of the magnet • This arrangement allows measurement of M/BH (ratio of magnetic moment to horizontal component of Earth's field) • The magnetic field due to the bar magnet is perpendicular to Earth's horizontal field in this position

Formula used: $\frac{M}{B_H} = \frac{4\pi}{\mu_0}\frac{(d^2-l^2)^2}{2d}\tan\theta$, where θ is the deflection angle

Techniques to eliminate reading errors in magnetic deflection angle measurements:

1. Symmetrical reading technique:

   • Read both ends of the compass pointer
   • Calculate the average of the two readings
   • Eliminates errors from pivot point eccentricity

2. Mirror alignment method:

   • Place a plane mirror beneath the compass scale
   • Position your eye so the pointer aligns with its reflection
   • Ensures perpendicular viewing angle to eliminate parallax

3. Rotation compensation:

   • Take readings, then rotate the compass 180°
   • Repeat measurements in the new orientation
   • Average all readings to cancel systematic errors

4. Multiple measurement protocol:

   • Vary the distance between magnet and compass
   • Plot results and identify outliers
   • Use statistical methods to determine best-fit values

Example application: When determining M/BH ratio (magnetic moment/horizontal field) using Tan-A position of Gauss, these techniques can improve measurement precision from ±5% to better than ±1%.

The linear relationship between cotθ and (d²-l²)²/2d directly verifies the inverse square law because:

• For Tan-A position: cotθ = (4π/μ₀)(BH/M)[(d²-l²)²/2d]
• When plotted, this produces a straight line through the origin
• This confirms that magnetic field B decreases with the square of distance (1/r²)
• The linearity demonstrates that the equation B = (μ₀/4π)(2M/r³) is valid
• The straight-line fit with zero y-intercept validates that no additional terms affect the relationship

This relationship is derived from the basic principle that magnetic field due to a magnetic pole is inversely proportional to the square of the distance from the pole.

Tan-A and Tan-B positions differ in their mathematical relationships:

• Tan-A position (end-on):

  • cotθ = (4π/μ₀)(BH/M)[(d²-l²)²/2d]
  • Plots cotθ vs. (d²-l²)²/2d

• Tan-B position (broadside-on):

  • cotθ = (4π/μ₀)(BH/M)(d²+l²)³/²
  • Plots cotθ vs. (d²+l²)³/²

Both yield valid results because:

1. Both derive from the same fundamental inverse square law
2. Both produce linear graphs through the origin
3. Each represents a different geometric configuration of the same physical system
4. The mathematical differences account for the different field orientations
5. The consistent linearity in both cases provides stronger validation than either alone

This demonstrates the robustness of the inverse square law regardless of measurement configuration.

The restoring torque on an oscillating magnetic needle in a uniform magnetic field is:

$\Gamma = -MBH\sin\theta$

Where:

• $M$ = magnetic moment of the needle
• $BH$ = horizontal component of the magnetic field
• $\theta$ = angular displacement from equilibrium

For small oscillations, $\sin\theta \approx \theta$, so: $\Gamma = -MBH\theta$

This creates simple harmonic motion because:

• The torque is proportional to displacement
• The torque is directed opposite to displacement
• The angular acceleration $\alpha = \Gamma/I = -(MBH/I)\theta = -\omega^2\theta$

This gives a time period: $T = 2\pi\sqrt{\frac{I}{MBH}}$

Application: This principle is used in oscillation magnetometers to determine $MBH$.

When a closed surface encloses only one pole of a dipole:

For magnetic dipoles:

• Field lines entering the surface must equal those exiting
• The net magnetic flux remains zero (∮B⃗·dS⃗=0)
• Field lines from the enclosed pole must curve back through the surface to reach the opposite pole
• This reflects the absence of magnetic monopoles

For electric dipoles:

• Field lines can terminate on or originate from charges
• The net electric flux equals the enclosed charge divided by ε₀ (∮E⃗·dS⃗=qenclosed/ε₀)
• If only a positive charge is enclosed, there is a net outward flux
• If only a negative charge is enclosed, there is a net inward flux

This fundamental difference demonstrates why isolated electric charges can exist while magnetic monopoles cannot.

Deflection Magnetometer Configurations

Tan-A Position (End-On)

• Geometric Arrangement:
  • Arms aligned east-west
  • Bar magnet placed parallel to arm (end-on to compass)
  • Creates field perpendicular to Earth's horizontal field ($B_H$)
• Field Equation: $B = \frac{\mu_0}{4\pi}\frac{2Md}{(d^2-l^2)^2}$
• Deflection Equation: $\frac{M}{B_H} = \frac{4\pi}{\mu_0}\frac{(d^2-l^2)^2}{2d}\tan\theta$
• Distance Dependence: Fourth power $(d^2-l^2)^2$ in denominator
• Field Strength: Stronger field effect

Tan-B Position (Broadside-On)

• Geometric Arrangement:
  • Arms aligned north-south
  • Bar magnet placed perpendicular to arm (broadside-on to compass)
  • Also creates field perpendicular to Earth's horizontal field
• Field Equation: $B = \frac{\mu_0}{4\pi}\frac{M}{(d^2+l^2)^{3/2}}$
• Deflection Equation: $\frac{M}{B_H} = \frac{4\pi}{\mu_0}(d^2+l^2)^{3/2}\tan\theta$
• Distance Dependence: 3/2 power $(d^2+l^2)^{3/2}$
• Field Strength: Generally weaker field effect

Common Underlying Principle

Both configurations rely on the tangent law of perpendicular fields:

• The compass needle aligns with the resultant of two perpendicular magnetic fields
• The tangent of deflection angle equals the ratio of perpendicular fields: $\tan\theta = \frac{B}{B_H}$
• This creates a "magnetic force balance" allowing precise measurement of $\frac{M}{B_H}$

Note: Both methods measure the same physical quantity but through different geometric arrangements, providing verification through multiple approaches.