The magnetic field strength due to a short bar magnet at a distance r from the midpoint of the magnet along its axial line is B. If the magnet is cut into two equal halves by a plane perpendicular to the magnetic axis then the magnetic field strength at a distance r from the midpoint along the axial line for any of the one-piece is equal to:

CONCEPT:

Magnetic dipole:

• A Magnetic Dipole consists of two unlike poles of equivalent strength and is separated by a small distance.

Magnetic moment:

• It is defined as the product of pole strength and the distance between the poles of a magnet.

​⇒ M = m × 2l

where M = magnetic moment, m = pole strength and 2l = distance between the poles

Magnetic field due to a bar magnet on the axial point:

• Magnetic field due to a bar magnet on the axial point at a distance r from the centre of magnet is given as,

\(\Rightarrow B=\frac{\mu_o}{4\pi}\frac{2Mr}{(r^2-l^2)^2}\)

If r >> l

\(\Rightarrow B=\frac{\mu_o}{4\pi}\frac{2M}{r^3}\)

CALCULATION:

Given r1 = r2 = r, B1 = B, m₁ = m2 = m, initial length of the bar magnet l1 = 2l, and final length of the bar magnet l₂ = l

Case 1:

• The magnetic moment for the magnet is given as,

​⇒ M₁ = m₁ × l₁

​⇒ M₁ = M = m × 2l

• We know that magnetic field due to a bar magnet on the axial point at a distance r from the center of magnet is given as,

\(\Rightarrow B_1=\frac{\mu_o}{4\pi}\frac{2M_1}{r_1^3}\)

\(\Rightarrow B_1=B=\frac{\mu_o}{4\pi}\frac{2M}{r^3}\)     -----(1)

Case 2: (When the magnet is cut into two equal halves by a plane perpendicular to the magnetic axis)

• The magnetic moment for each piece of the magnet is given as,

​⇒ M2 = m2 × l2

​⇒ M2 = m×l

\(\Rightarrow M_2=\frac{M}{2}\)

• So the magnetic field strength at a distance r from the midpoint along the axial line for any of the one-pieces is given as,

\(\Rightarrow B_2=\frac{\mu_o}{4\pi}\frac{2M_2}{r_2^3}\)

\(\Rightarrow B_2=\frac{1}{2}\times\frac{\mu_o}{4\pi}\frac{2M}{r^3}\)

\(\Rightarrow B_2=\frac{B}{2}\)

• Hence, option 3 is correct.