Magnetism Concept Page - 13

Definition

Magnetic susceptibility

For a linear material, M=χH$M=\chi H$ where χ$\chi$ is called the magnetic susceptibility of the material.
The magnetic susceptibility of a paramagnetic substance is 3×10−4$3\times {10}^{-4}$ .It is in a magnetising field of 4×10−4$4\times {10}^{-4}$  amp/m. Find the magnetization of the substance.Magnetization M$M$ is related to magnetic susceptibility by :
M=χH$M=\chi H$
∴M=3×10−4×4×10−4=12×10−8$\therefore M=3\times {10}^{-4}\times 4\times {10}^{-4}=12\times {10}^{-8}$ A/m

Formula

Relative permeability and magnetic susceptibility

μr=1+χ${\mu }_r=1+\chi$
where μr${\mu }_r$ is the relative permeability and
χ$\chi$ is the magnetic susceptibility

Example

Magnetic moment and pole strength of a permanent magnet

Magnetic moment of bar magnet:$Magneticmomentofbarmagnet:$
The bar magnet possesses a magnetic moment which implies that how bar magnet will align itself when placed in an external magnetic field. Its direction is from south pole to north pole.
It is analogous to electric dipole moment where we replace electric charge with magnetic pole strength.
Hence, mathematically, M⃗ =m(l)$\overrightarrow{M}=m(l)$
where, m = pole strength
            d = length of bar magnet

Example:$Example:$
The magnetic moment of a bar magnet is 3.6x10−3${}^{-3}$ A.m2${}^2$. Its pole strength is 120 milli amp.m. Find its magnetic length.
Magnetic  length = Magnetic  Moment/Pole  Strength = 3.6×10−3/(120×10−3)=0.03m.=3cm.$3.6\times {10}^{-3}/(120\times {10}^{-3})=0.03m.=3cm.$

Definition

Bohr magneton

The Bohr magneton μB${\mu }_B$ is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum.μB=eℏ2me${\mu }_B={\displaystyle \frac{e\hslash }{2m_e}}$
where e$e$ is the elementary charge, ℏ$\hslash$ is the reduced Planck's constant, me$m_e$ is the electron rest mass.
The value of Bohr magneton in SI units is 9.27400968(20)×10−24JT−1$9.27400968(20)\times {10}^{-24}J{T}^{-1}$

Definition

Permeability and permittivity of free space

"Permittivity is concerned with electric fields and is the "ability of a material to polarize in response to the (electric) field".ϵo${ϵ}_o$
"Permeability is concerned with magnetic fields and is "the degree of magnetization of a material in response to a magnetic field" μo${\mu }_o$There is an interesting relation between ϵ0${ϵ}_0$the permittivity of free space; μ0${\mu }_0$the permeability of free space; and c$c$, the speed of light in vaccum :
ϵ0μ0=(4πϵ0)(μ04π)=19×109(10−7)=1(3×108)2=1c2${ϵ}_0{\mu }_0=(4\pi {ϵ}_0)({\displaystyle \frac{{\mu }_0}{4\pi }})={\displaystyle \frac{1}{9\times {10}^9}}({10}^{-7})={\displaystyle \frac{1}{{(3\times {10}^8)}^2}}={\displaystyle \frac{1}{c^2}}$

Example

Energy density in a magnetic field

A circular loop of wire 4$4$ cm in radius carries a current of 80$80$ A. Find the energy density at the centre of the loop.Magnetic field at the center of the circle B=μ0i2r$B={\displaystyle \frac{{\mu }_{0}i}{2r}}$
Energy density u=B22μ0=(μ0i2r)22μ0=μ0i28r2=4π×10−7×8028×(4×10−2)2=0.2πJ/m3$u={\displaystyle \frac{B^2}{2{\mu }_0}}={\displaystyle \frac{{\left({\displaystyle \frac{{\mu }_{0}i}{2r}}\right)}^2}{2{\mu }_0}}={\displaystyle \frac{{\mu }_0{i}^2}{8r^2}}={\displaystyle \frac{4\pi \times {10}^{-7}\times {80}^2}{8\times (4\times {10}^{-2}{)}^2}}=0.2\pi J/m^3$

Example

Torque balance in magnetic field

A magnetic needle suspended parallel to a magnetic field requires 3√$\sqrt{3}$ J of work to turn it through 60o${}^o$. Find the torque needed to maintain the needle in this position.
According to work energy theorem
W=Ufinal−Uinitial=MB(cos0−cos60o)$W=U_{final}-U_{initial}=MB(cos0-cos{60}^o)$
W=MB2=3√ J${\displaystyle W=\frac{MB}{2}=\sqrt{3}J}$     .......... (i)
τ=M⃗ ×B⃗ =MBsin60o=(MB3√2)$\tau =\overrightarrow{M}\times \overrightarrow{B}=MBsin{60}^o={\displaystyle \left(\frac{MB\sqrt{3}}{2}\right)}$ .......... (ii)
From equation (i) and (ii)
τ=23√×3√2=3J${\displaystyle \tau =\frac{2\sqrt{3}\times \sqrt{3}}{2}=3J}$

Example

Problems on changing magnetic moment

The magnetic moment of a bar magnet is M. If it is cut into two pieces in the ratio 1:2 perpendicular to its length.Find the ratio of their magnetic moments.Magnetic moment of a magnet=ml$=ml$
where m=$m=$pole strength
l=$l=$effective  distance  between  the  two  poles
When the magnet is cut, the pole strength(m) remains the same.
Therefore the ratio of Magnetic Moments is the same as the ratio of their lengths.
Therefore required ratio is 1:2$1:2$

Example

SHM of a bar magnet in a uniform magnetic field

A freely suspended short bar magnet makes 20 oscillations per minute at a place where earths horizontal field is 40μ$\mu$ T. Another short bar magnet of moment 1.6 A-m2${}^2$ is placed at 20 cm east pointing its north pole to north. Find the number of oscillations per minute (nearly).The time period of oscillation is given by 
T=2πIMB−−−−√$T=2\pi \sqrt{{\displaystyle \frac{I}{MB}}}$
The magnetic field due to the bar magnet on an equatorial line of a short magnet is given by
B=μo4πMd3=10−7×1.6(20×10−2)3$B={\displaystyle \frac{{\mu }_o}{4\pi }}{\displaystyle \frac{M}{d^3}}={10}^{-7}\times {\displaystyle \frac{1.6}{(20\times {10}^{-2}{)}^3}}$ 
B=20μ$B=20\mu$ T
f2f1=B2B1−−−√${\displaystyle \frac{f_2}{f_1}}=\sqrt{{\displaystyle \frac{B_2}{B_1}}}$

B2=40μT−20μT=20μT$B_2=40\mu T-20\mu T=20\mu T$
f2=200−−−√$f_2=\sqrt{200}$ oscillations per minute
f2≈14$f_2\approx 14$ oscillations per minute

Definition

Construction of a moving coil galvanometer

The galvanometer consists of a coil, with many turns, free to rotate about a fixed axis, in a uniform radial magnetic field. There is a cylindrical soft iron core which not only makes the field radial but also increases the strength of the magnetic field.When a current flows through the coil, a torque acts on it.The magnetic torque tends to rotate the coil. A spring provides a counter torque that balances the magnetic torque, resulting in a steady angular deflection. The deflection is indicated on the scale by a pointer attached to the spring.

BookMarks

Page 1  Page 2  Page 3  Page 4  Page 5  Page 6  Page 7  Page 8  Page 9  Page 10  Page 11
Page 12  Page 13  Page 14  Page 15  Page 16