Question

The given equation represents a magnetic field strength \( \vec{H}(r, \theta, \phi) \) in the spherical coordinate system, in free space. Here, \( \hat{r} \) and \( \hat{\theta} \) represent the unit vectors along \( r \) and \( \theta \), respectively. The value of \( P \) in the equation should be (rounded off to the nearest integer). \[ \vec{H}(r, \theta, \phi) = \frac{1}{r^3} \big( \hat{r} P \cos \theta + \hat{\theta} \sin \theta \big) \]

Hint:

For spherical coordinate systems, the magnetic field of a dipole follows the pattern \( \frac{1}{r^3} (2 \hat{r} \cos \theta + \hat{\theta} \sin \theta) \). Use this standard form for comparisons.

Answer 1

Step 1: Use the governing equation for the magnetic dipole field. The magnetic field \( \vec{H} \) in free space for a magnetic dipole is given by: \[ \vec{H}(r, \theta, \phi) = \frac{1}{r^3} \big( 2 \hat{r} \cos \theta + \hat{\theta} \sin \theta \big). \] Step 2: Compare the given equation with the standard form. From the given equation: \[ \vec{H}(r, \theta, \phi) = \frac{1}{r^3} \big( \hat{r} P \cos \theta + \hat{\theta} \sin \theta \big), \] it is clear that \( P = 2 \) by comparison with the standard equation for a magnetic dipole field.