MEC O10

Analysis: We are asked to find the magnetic induction vectors at two specific points due to a magnetic dipole $\vec{m}$ located at the origin. Based on the standard convention for such problems and the provided options, the dipole moment $\vec{m}$ is oriented along the positive y-axis, i.e., $\vec{m} = m\hat{j}$.

We need to evaluate the magnetic field $\vec{B}$ at:

• Point A: $(0, y)$ on the y-axis (Axial position or “End-on” position).
• Point B: $(x, 0)$ on the x-axis (Equatorial position or “Broadside-on” position).

1. Magnetic Field at Point A (Axial Point)

Point A lies on the axis of the dipole. For a point on the axis at distance $y$, the magnetic field vector is parallel to the dipole moment.

Formula: $|\vec{B}_{axial}| = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$

Since $\vec{m}$ is in the $+\hat{j}$ direction:

$$ \vec{B}_A = \frac{\mu_0}{4\pi} \frac{2m}{y^3} \hat{j} = \frac{\mu_0 m}{2\pi y^3} \hat{j} $$

2. Magnetic Field at Point B (Equatorial Point)

Point B lies on the equatorial line (perpendicular bisector) of the dipole. For an equatorial point at distance $x$, the magnetic field vector is antiparallel to the dipole moment.

Formula: $|\vec{B}_{eq}| = \frac{\mu_0}{4\pi} \frac{m}{r^3}$

Since $\vec{m}$ is along $+\hat{j}$, the field will be along $-\hat{j}$:

$$ \vec{B}_B = -\frac{\mu_0}{4\pi} \frac{m}{x^3} \hat{j} $$