Why do magnetic field lines not intersect each other?

If magnetic field lines intersect each other, then at the intersection point there will be two directions of the same field which is not possible. Hence the field lines do not cross or intersect each other.

The magnetic field in a given region is uniform. Draw a diagram to represent it.

Uniform magnetic field is represented by parallel magnetic field lines in same direction.

Derive an expression for magnetic field at a point on the axial line of a bar magnet.

Let \$NS\$ be a bar magnet of magnetic length \$2l\$ and having each pole of magnetic strength \$m\$ . O is the center of magnet and \$P\$ is a point on axial line at a distance \$r\$ from the center \$O\$ of magnet , at which magnetic field has to be measured .The magnetic field \$B_{1}\$ at \$P\$ due to \$N\$ pole of magnet , \$B_{1}=\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{NP^{2}}\$or \$B_{1}=\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{(r-l)^{2}}\$ (along PX) .......................eq1And , the magnetic field \$B_{2}\$ at \$P\$ due to \$S\$ pole of magnet , \$B_{2}=\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{SP^{2}}\$or \$B_{2}=\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{(r+l)^{2}}\$ (along PS) ......................eq2Therefore , resultant magnetic field at point \$P\$ , \$B=B_{1}-B_{2}\$ (-ive sign is due to opposite directionS of \$B_{1}\$ and \$B_{2}\$)It is clear from eq1 and eq2 that \$B_{1}>B_{2}\$ ,therefore the direction of \$B\$ will be along \$PX\$ .or \$B=\$\$\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{(r-l)^{2}}\$\$-\dfrac{\mu_{0}}{4\pi}.\dfrac{m}{(r+l)^{2}}\$ (alongPX)or \$B=\dfrac{\mu_{0}}{4\pi}.\dfrac{m(4rl)}{(r^{2}-l^{2})^{2}}\$ (alongPX)Now , \$m(2l)=M\$ (magnetic dipole moment of magnet)Hence , \$B=\dfrac{\mu_{0}}{4\pi}.\dfrac{2Mr}{(r^{2}-l^{2})^{2}}\$ (alongPX)

The ratio of magnitude of electrostatic force and gravitational force for an electron and proton is

Here, for an electron and a protonElectrostatic force, \$\left| { F }_{ e } \right| =\cfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \cfrac { { e }^{ 2 } }{ { r }^{ 2 } } \$Gravitational force, \$\left| { F }_{ g } \right| =\cfrac { G{ m }_{ e }{ m }_{ p } }{ { r }^{ 2 } } \$\$\therefore \cfrac { \left| { F }_{ e } \right| }{ \left| { F }_{ g } \right| } =\cfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \cfrac { { e }^{ 2 } }{ G{ m }_{ e }{ m } } =\cfrac { 9\times { 10 }^{ 9 }\times { \left( 1.6\times { 10 }^{ -19 } \right) }^{ 2 } }{ 6.67\times { 10 }^{ -11 }\times 9\times { 10 }^{ -31 }\times 1.66\times { 10 }^{ -27 } } =2.4\times { 10 }^{ 39 }\$

\$F_g\$ and \$F_e\$ represent gravitational and electrostatic force respectively between electrons situated at a distance \$0.1\,m. F_g/F_e\$ is of the order

\${ F }_{ y }\$ and \${ F }_{ e }\$ represent gravitational and electrostatic force respectively between electrons situated at a distance \$=0.1m\$let, us take the ratio\$\dfrac { { F }_{ e } }{ { F }_{ g } } =\dfrac { \dfrac { 1 }{ 4\pi { \epsilon }_{ 0 } } \dfrac { { q }^{ 2 } }{ { d }^{ 2 } } }{ G\dfrac { { m }^{ 2 } }{ { d }^{ 2 } } } \$\$\dfrac { { F }_{ e } }{ { F }_{ g } } =\dfrac { 1 }{ 4\pi { \epsilon }_{ 0 } } \times \dfrac { 1 }{ G } \times \dfrac { { q }^{ 2 } }{ { m }^{ 2 } } \$\$\dfrac { { F }_{ e } }{ { F }_{ g } } =9\times { 10 }^{ 9 }\times \dfrac { 1 }{ 6.67\times { 10 }^{ -11 } } \times \dfrac { { \left( 1.6\times { 10 }^{ -19 } \right) }^{ 2 } }{ { \left( 9.1\times { 10 }^{ -31 } \right) }^{ 2 } } \$\$\Rightarrow \dfrac { { F }_{ e } }{ { F }_{ g } } =4.17\times { 10 }^{ 42 }\$or, \$\dfrac { { F }_{ e } }{ { F }_{ g } } \$ is of the order of \${ 10 }^{ -42 }\$.

Check that the ratio \$ke^2\$ /G \$m_e\,m_p\$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

The given ratio is \$\displaystyle \frac{ke^2}{Gm_cm_p}\$Where,\$G =\$ Gravitational constantIts unit is \$N\,m^2kg^{-2}\$.\$m_e\$ and \$m_p=\$ Masses of electron and proton.Their unit is kg.\$e =\$ Electric charge.Its unit is C.\$k = A\$ constant \$\displaystyle =\frac{1}{4\pi\in_0}\$\$\in_0=\$ Permittivity of free spaceIts unit is \$N\,m^2C^{-2}\$. Unit of the given ratio \$\displaystyle \frac{ke^2}{Gm_cm_p}=\frac{[Nm^2C^2][C^{-2}]}{[Nm^2\,kg^{-2}][kg][kg]}\$ \$=M^0L^0T^0\$Hence, the given ratio is dimensionless.\$e=1.6\times 10^{-19}C\$\$G=6.67\times 10^{-11}N\,m^2kg^{-2}\$\$m_e=9.1\times 106{-31}kg\$\$m_p=1.66\times 10^{-27}kg\$\$\displaystyle \frac{ke^2}{Gm_cm_p} = \frac{9\times 10^9\times(1.6\times 10^{-19})^2}{6.67\times 10^{-11}\times 9.1\times 10^{-3}\times 1.67\times 10^{-22}}\approx 2.3\times 10^{39}\$This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.

The ratio of electric force to the gravitational force between two protons separated by a finite distance is of the order of:

\$Gravitational\ force\ = \dfrac{G(m_p)^2}{r^2}\$\$Electric\ force\ = \dfrac{K(e)(e)}{r^2}\$\$Ratio\ = \dfrac{K(1.6\times 10^{-19})^2}{r^2}\times \dfrac{r^2}{G\times (1.6\ 7\times 10^{-27})^2}\$\$= \dfrac{(9\times 10^9)\times (1.6\times 10^{-19})^2}{(6.6\ 7\times 10^{-11})(1.6\ 7\times 10^{-27})^2}\$\$=1.23\times 10^{20}\times 10^{16}\$\$= 10^{36}\$(order of magnitude)

Identify the poles of the magnet in the figure (a) and (b) shown below.

The magnetic field lines emerge from north pole and merge at the south pole

What are magnetic field lines? How is the direction of magnetic field at a point in a magnetic field determined using field lines?

A magnetic field line is the path along which a free north-pole tends to move. The direction of a magnetic field at a point is determined by placing a small compass needle. The N - pole of compass indicates the direction of magnetic field at that point.

Draw a diagram to show the magnetic fields lines around a bar magnet.

Magnetic fields lines around a bar magnet.

Properties of Magnetic Field Lines Around a Bar Magnet

result

Magnetic lines of bar magnet

Magnetic lines of force of a magnet shows the direction in which a magnetic material will be aligned when placed at the location. Magnetic field lines for a bar magnet are as shown in the diagram.

definition

Properties - Magnetic lines of force

Properties of magnetic lines of force are:

They are continuous closed curves.
They originate from the North Pole and terminate at the South Pole.
Two lines do not intersect each other.
They are crowded where the magnetic field is strong, i.e. near the poles.

definition

Magnetic field follows superposition principle

The magnetic field of several sources is the vector addition of magnetic field of each individual source. Thus it follows the superposition principle.

definition

Properties of magnetic fields of lines

Magnetic field lines are imaginary lines that represent magnetic fields. It gives direction of magnetic force. An advantage of using magnetic field lines as a representation is that laws of magnetism and (electromagnetism) can be stated using concepts like 'number' of field lines through a surface.

Magnetic field lines are like streamlines in fluid flow, they represent something continuous, and a different resolution would show more or fewer lines. Various phenomena have the effect of displaying magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field, form lines that correspond to field lines. Field lines can be used as a qualitative tool to visualize magnetic forces.' Unlike' poles of magnets attract because they are linked by many field lines, 'like' poles repel because their field lines do not meet, but run parallel, pushing on each other.

definition

The net magnetic flux through a closed surface is zero

The contribution to magnetic flux for a given area is equal to the area times the component of magnetic field perpendicular to the area.
For a closed surface the number of magnetic field lines entering the surface is equal to the number of field lines exiting it.Thus the sum of magnetic flux is always equal to zero.

definition

Magnetic Field strength due to bar magnet

Consider a bar magnet of length 2l and pole strength m. Suppose a point P on the axis of the magnet at a distance r from its center. (r-l) is the distance of P form the N-pole of the magnet. The magnetic field intensity at P due to the north-pole of the magnet is

B_{1} = \frac{μ _{o} m}{4 π r ^{2}}

B_{1} = \frac{μ _{o} m}{4 π ( r - l ) ^{2}}

which is directly away from N-pole. Since the south of the magnet is at a distance r + l from P, so magnetic field intensity at P due to S-pole is

B_{2} = \frac{μ _{o} m}{4 π ( r + l ) ^{2}}

which is direct towards, the S-pole of the magnet. The magnetic field intensity B at P is the resultant of these two fields

B = B_{1} + (- B_{2})

B = B_{1} - B_{2})

= \frac{μ _{o} m}{4 π} \times [\frac{1}{( r - l ) ^{2}} - \frac{1}{( r + l ) ^{2}}]

= \frac{μ _{o} m}{4 π} \times \frac{4 l r}{( r ^{2} - l ^{2} ) ^{2}}

$= \frac{μ _{o} 2 M d}{4 π ( r ^{2} - l ^{2} ) ^{2}}$

example

Example - Magnetic filed strength due to a bar magnet

A magnet is

10

cm long and its pole strength is

120

CGS units (

1

CGS unit of pole strength=

0.1 A m

). Find the magnitude of the magnetic field

B

at a point on its axis at a distance 20 cm from it.
Solution:
The pole strength is

m = 120

CGS units =

12 A m

.
Magnetic field is

2 l = 10 c m

l = 0.05 m

.
Distance from the magnet is

d = 20 c m = 0.2 m

.
The field

B

at a point in end-on position is:

B = \frac{μ _{0}}{4 π} \frac{2 M d}{( d ^{2} - l ^{2} ) ^{2}} = \frac{μ _{0}}{4 π} \frac{2 m l d}{( d ^{2} - l ^{2} ) ^{2}} = (10^{- 7} \frac{T m}{A}) \frac{4 \times ( 1 2 A m ) \times ( 0 . 0 5 m ) \times ( 0 . 2 m )}{[ ( 0 . 2 m ) ^{2} - ( 0 . 0 5 m ) ^{2} ] ^{2}} = 3.4 \times 10^{- 5} T

Properties of Magnetic Field Lines Around a Bar Magnet

result

Magnetic lines of bar magnet

definition

Properties - Magnetic lines of force

definition

Magnetic field follows superposition principle

definition

Properties of magnetic fields of lines

definition

The net magnetic flux through a closed surface is zero

definition

Magnetic Field strength due to bar magnet

example

Example - Magnetic filed strength due to a bar magnet

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