A current loop and a magnetic dipole model are equivalent representations:

1. Current Loop: Physical circular current producing magnetic field described by $\vec{B} = \frac{\mu_0}{4\pi}\frac{2\vec{\mu}}{d^3}$ at axial points, where $\vec{\mu}=i\vec{A}$ is the magnetic moment.

2. Magnetic Dipole Model: Hypothetical model with:

   • Positive (north) and negative (south) magnetic charges separated by distance d
   • Magnetic dipole moment $\vec{\mu}=md$ points from south to north pole
   • Same resulting field equations as current loop when $md = iA$

The equivalence allows us to replace complex current distributions with simpler dipole representations in calculations.

Using the lever arm approach:

$\tau = 2mB(l\sin\theta) = 2mlB\sin\theta = MB\sin\theta$

Where:

• m is the pole strength of each pole
• 2l is the magnetic length (distance between poles)
• l·sin θ is the lever arm for each force
• 2ml = M is the magnetic moment of the bar magnet

The factor of 2 appears because both poles contribute to the torque:

• The north pole experiences force mB upward
• The south pole experiences force mB downward
• Each force acts at distance l·sin θ from the axis of rotation

This approach demonstrates that the torque depends on both the strength of the magnetic interaction (mB) and the geometric arrangement (l·sin θ).

The magnetic scalar potential at point P due to a magnetic dipole is:

$V = \frac{\mu_0}{4\pi}\frac{M\cos\theta}{r^2}$

Where:

• $M$ is the magnetic moment of the dipole ($M = 2ml$ where $m$ is pole strength)
• $r$ is the distance from center O to point P
• $\theta$ is the angle between the dipole axis and the line OP
• $\mu_0$ is the permeability of free space

This potential decreases with the square of the distance and varies with the cosine of the angle, being maximum along the dipole axis and zero perpendicular to it.

For a magnetic dipole with poles S and N:

1. The exact calculation requires finding the field contribution from each pole:

   • From S pole: $B_S = -\frac{\mu_0}{4\pi}\frac{m}{PS^2}$ (toward the pole)
   • From N pole: $B_N = \frac{\mu_0}{4\pi}\frac{m}{PN^2}$ (away from the pole)

2. For a point P far from the dipole (r ≫ l, where 2l is the dipole length):

   • The distances can be approximated as:
     • PS ≈ r + l·cosθ
     • PN ≈ r - l·cosθ

3. Using these approximations and mathematical simplification leads to the dipole field formula: $B = \frac{\mu_0}{4\pi}\frac{M}{r^3}\sqrt{1+3\cos^2\theta}$

This approximation is valid when the observation point is significantly farther from the dipole than the dipole's length.

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.

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.

The cylindrical pole design ensures that the magnetic field lines remain parallel to the plane of the coil throughout its rotation. This geometric arrangement:

• Maintains a constant magnetic field strength at the arms of the coil
• Ensures the deflecting torque equation Γ = niAB remains valid at all deflection angles
• Prevents torque variation that would occur with flat-faced poles
• Creates a radial magnetic field pattern that keeps the force perpendicular to the coil arms

This design is critical for maintaining linear proportionality between current and deflection angle (i = k·θ/nAB), making the galvanometer readings accurate and consistent.

Taking multiple readings by rotating and flipping the bar magnet addresses:

• Non-coincidence of magnetic and geometric centers: The magnetic axes can be offset from the physical center
• Pole asymmetry: North and south poles may have slightly different magnetic strengths

The specific technique involves:

1. Taking readings with the magnet in its original orientation (distance d₁)
2. Rotating the magnet 180° about a vertical axis (swapping N and S positions)
3. Taking readings in this reversed orientation (distance d₂)
4. Averaging all measurements

This methodology:

• Neutralizes systematic errors from magnetic center displacement
• Improves accuracy when calculating M/BH (ratio of magnetic moment to horizontal component of Earth's field)
• Creates a more precise determination of the actual magnetic field at the measurement point

Without this technique, calculated magnetic moments would contain systematic errors proportional to the offset between centers.

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.

Magnetic Field Contributions from Bar Magnets

The total magnetic field from a bar magnet (magnetic moment $M = 2ml$) results from vector addition of fields from both poles:

End-On Position (Axial Point)

• North pole contribution: $B_N = \frac{\mu_0 m}{4\pi(d-l)^2}$ (directed away from magnet)
• South pole contribution: $B_S = \frac{\mu_0 m}{4\pi(d+l)^2}$ (directed toward magnet)
• Total field: Fields have opposite directions along same axis, yielding: $B = B_N - B_S = \frac{\mu_0}{4\pi}\left[\frac{m}{(d-l)^2} - \frac{m}{(d+l)^2}\right] = \frac{\mu_0}{4\pi}\frac{2Md}{(d^2-l^2)^2}$
• Field points away from the magnet

Broadside-On Position (Perpendicular Bisector)

• Both poles produce fields according to: $\vec{B}_{\text{pole}} = \frac{\mu_0}{4\pi} \frac{m \cdot \vec{r}}{r^3}$
• Poles are equidistant from observation point, so field magnitudes are equal
• Vector addition: Horizontal components cancel; vertical components add constructively
• Total field: Parallel to magnet's axis with magnitude: $B = \frac{\mu_0}{4\pi} \frac{M}{(d^2 + l^2)^{3/2}}$

Note: This field pattern is analogous to that of a current loop, despite no actual current flow in the magnet.