i. Calculate the angles between position vectors of P and Q with the direction of m.

Question

At a given place on the Earth, a bar magnet of magnetic moment \(\overset\rightarrow{m}\) is kept horizontal in the East-West direction. P and Q are the two neutral points due to magnetic field of this magnet and \(\overset\rightarrow{B}_H\) is the horizontal component of the Earth’s magnetic field.

i. Calculate the angles between position vectors of P and Q with the direction of \(\overset\rightarrow{m}.\)

ii. Points P and Q are 1 m from the centre of the bar magnet and B_H = 3.5 × 10^-5 T.

Calculate magnetic dipole moment of the bar magnet.

Neutral point is that point where the resultant magnetic field is zero.

Answer 1

 i. The direction of magnetic field \(\overset\rightarrow{B}\) due to the bar magnet is opposite to \(\overset\rightarrow{B}_H\) at the neutral points P and Q such that (θ + α) = 90° at P and (θ + α) = 270° at Question

∴ tan α = \(\frac{1}{2}\) tan θ

∴ tan θ = 2 tan α

= 2 tan (90 – θ) and 2 tan (270 – θ)

∴ tan θ = ± 2 cot θ

∴ tan²θ = 2 …….. (1)

∴ tanθ = ±√2

∴ θ = tan^-1 (±√2)

∴ θ = 54°44′ and 180° – 54° 44° = 125°16′

ii. For magnetic dipole moment of the bar magnet:

From equation (2), tan² θ = 2

∴ sec² θ = 1 + tan² θ = 1 + 2 = 3

∴ cos² θ = \(\frac{1}{3}\)

r = 1 m and B = B_H = 3.5 × 10^-5 T ……. (Given)

we have,