The given equation represents a magnetic field strength $H(r, \theta, \phi)$ 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). $\bar H (r, \theta, \phi) $ = $\frac{1}{r^3}(\hat r P cos \theta + \hat \theta sin \theta)$
Magnetic Field Strength Equation Analysis
The problem presents an equation for the magnetic field strength $H$ in spherical coordinates:
$ \vec{H}(r, \theta, \phi) = \frac{1}{r^3}(\hat r P \cos \theta + \hat \theta \sin \theta) $This mathematical structure suggests that the equation represents the magnetic field generated by a magnetic dipole source.
Standard Magnetic Dipole Field Equation
The established formula for the magnetic field strength $\vec{H}$ resulting from a magnetic dipole moment $\vec{m}$, when the moment is aligned along the z-axis and expressed in spherical coordinates $(r, \theta, \phi)$, is:
$ \vec{H}(r, \theta) = \frac{m}{4\pi r^3} (2 \cos \theta \hat{r} + \sin \theta \hat{\theta}) $In this standard equation, $m$ denotes the magnitude of the dipole moment $|\vec{m}|$. The field strength depends only on the radial distance $r$ and the polar angle $\theta$, not the azimuthal angle $\phi$. The equation can be rewritten to better match the given format:
$ \vec{H}(r, \theta) = \frac{1}{r^3} \left( \frac{m}{4\pi} (2 \cos \theta \hat{r} + \sin \theta \hat{\theta}) \right) $Comparing Field Equations to Find P
To determine the value of the parameter $P$, we directly compare the provided equation with the standard magnetic dipole field equation:
- Provided Equation: $ \vec{H} = \frac{1}{r^3}(\hat r P \cos \theta + \hat \theta \sin \theta) $
- Standard Dipole Equation: $ \vec{H} = \frac{1}{r^3} \left( \frac{m}{4\pi} (2 \cos \theta \hat{r} + \sin \theta \hat{\theta}) \right) $
By aligning the terms, we equate the coefficients associated with the unit vectors $\hat{r}$ and $\hat{\theta}$ within the parentheses:
- Equating the coefficients for the $\hat{\theta}$ component: $\sin \theta = \frac{m}{4\pi} \sin \theta$. This comparison yields the relationship $\frac{m}{4\pi} = 1$.
- Equating the coefficients for the $\hat{r}$ component: $P \cos \theta = \frac{m}{4\pi} (2 \cos \theta)$.
Determining the Value of P
Substitute the result $\frac{m}{4\pi} = 1$ from the $\hat{\theta}$ component comparison into the equation derived from the $\hat{r}$ components:
$ P \cos \theta = (1) \times (2 \cos \theta) $This simplifies to:
$ P \cos \theta = 2 \cos \theta $For values of $\theta$ where $\cos \theta \neq 0$ (i.e., excluding the magnetic poles where $\theta = 0$ or $\theta = \pi$), we can divide both sides by $\cos \theta$:
$ P = 2 $The calculated value for $P$ is 2. When rounded to the nearest integer, the value remains 2.
Important Questions from Magnetostatics
A magnetic pressure which sets up or tends to set up flux in a magnetic circuit is called-
A coil of 600 turns and of resistance of 20 Ω is wound uniformly over a steel ring of mean circumference 30 cm and cross sectional area 9 cm2. If the relative permeability of the ring is 1600. Find the value of reluctance.
The unit of magnetic flux density is
The B-H curve for ______ will be a straight line passing through the origin.
The SI unit of permeability is:
