When an electric charge q is moving with velocity \(\vec { v } \) in the magnetic field \(\vec { B } \), it experiences a force, called magnetic force \(\vec { { F }_{ m } } \). After careful experiments, Lorentz deduced the force experienced by a moving charge in the magnetic field \(\vec { { F }_{ m } } \).
\(\vec { { F }_{ m } } =q(\vec { v } \times \vec { B } )\)             ...........(1)
In magnitude, F_m = qvB sinθ         .......(2)
The equations (1) and equation (2) imply
(i) \(\vec { { F }_{ m } } \) is directly proportional to the magnetic field \(\vec { B } \).
(ii) \(\vec { { F }_{ m } } \) is directly proportional to the velocity \(\vec { v } \).
(iii) \(\vec { { F }_{ m } } \) is directly proportional to sine of the angle between the velocity and magnetic field.
(iv) \(\vec { { F }_{ m } } \) is directly proportional to the magnitude of the charge q.
(v) The-direction of \(\vec { { F }_{ m } } \) is always perpendicular to \(\vec { v } \) and B as \(\vec { { F }_{ m } } \) in the cross product of \(\vec { v } \) and \(\vec { B } \).

(vi) The direction of \(\vec { { F }_{ m } } \) on a negative charge is opposite to the direction of \(\vec { { F }_{ m } } \) on positive charge provided other factors are identified as shown in Figure.
(vii) If the velocity \(\vec { v } \) of the charge, q is along the magnetic field \(\vec { B } \) then, \(\vec { { F }_{ m } } \) is zero.

The magnetic field from the center of a circular loop of radius R along the axis is given by
\(\vec { B } =\frac { { \mu }_{ 0 }I }{ 2 } \frac { { R }^{ 2 } }{ ({ R }^{ 2 }+{ z }^{ 2 })^{ \frac { 3 }{ 2 } } } \hat { k } \)
At larger distance z >> R, therefore R² + z² ≈ z²,
we have
\(\vec { B } =\frac { { \mu }_{ 0 }I }{ 2 } \frac { { R }^{ 2 } }{ { z }^{ 3 } } \hat { k } \)               ........(1)
Let A be the area of the circular loop A = πR². So rewriting the equation (1) in terms of the area of the loop, we have
\(\vec { B } =\frac { { \mu }_{ 0 }I }{ 4\pi } \frac { { R }^{ 2 } }{ { z }^{ 3 } } \hat { k } \)
\(\vec { B } =\frac { { \mu }_{ 0 } }{ 4\pi } \frac { 2IA }{ { z }^{ 3 } } \hat { k } \)              .......(2)
Comparing equation (2) with equation (1) dimensionally, we get
p_m = IA
where P_m is called a magnetic dipole moment. In vector notation,
\(\vec { { p }_{ m } } =I\vec { A } \)       .........(3)
This implies that a current - carrying circular loop behaves as a magnetic dipole of the magnetic moment \(\vec { { p }_{ m } } \) So, the magnetic dipole moment of any current loop is equal to the product of the current and area of the loop.

(i) Magnetic field lines are continuous closed curves. The direction of magnetic field lines is from the North pole to the South pole outside the magnet and South pole to the North pole inside the magnet.
(ii) The direction of a magnetic field at any point on the curve is known by drawing a tangent to the magnetic line of force at that point.
(iii) Magnetic field lines never intersect each other.
(iv) The degree of closeness of the field lines determines the relative strength of the magnetic field. The magnetic field is strong where magnetic field lines crowd and weak where magnetic field lines thin out.

(i) A freely suspended bar magnet will always point along the north-south direction.
(ii) A magnet attracts another magnet or magnetic substances towards itself. The attractive force is maximum near the end of the bar magnet. When a bar magnet is dipped into iron filling, they cling to the ends of the magnet.
(iii) When a magnet is broken into pieces, each piece behaves like a magnet with poles at its ends.
(iv) Two poles of magnet have pole strength equal to one another.
(v) The length of the bar magnet is called geometrical length and the length between two magnetic poles in a bar magnet is called magnetic length. Magnetic length is always slightly smaller than geometrical length.

(i) Magnets are classified into natural magnets and artificial magnets.
(ii) For example, iron, cobalt, nickel, etc. are natural magnets.
(iii) Strengths of natural magnets are very weak and the shapes of the magnet are irregular.
(iv) Artificial magnets are made by us in order to have desired shape and strength.
(v) If the magnet is in the form of rectangular shape or cylindrical shape, then it is known as bar magnet.

(i) Horizontal component:
The Earth's magnetic field is parallel to the surface of the Earth (i.e., horizontal) which implies that the needle of the magnetic compass rests horizontally at an angle of dip, I = 0⁰ as shown in the figure.
B_H = B_E
B_v = 0
This implies that the horizontal component is maximum at the equator and the vertical component is zero at the equator.

(ii) Vertical component: The Earth's magnetic field is perpendicular to the surface of the Earth (i.e., vertical) which implies that the needle of magnetic compass rests vertically at an angle of dip, I = 90° as shown in Figure
Hence,
B_H = 0
B_v = B_E
This implies that the vertical component is maximum at poles and the horizontal component is zero at poles.

It is defined as the deflection produced per unit voltage applied across the galvanometer.
V_s =\(\frac { \theta }{ V } \)
V_s = \(\frac { \theta }{ IR_{ g } } =\frac { NAB }{ KR_{ g } } =V_{ s }=\frac { 1 }{ GR_{ g } } =\frac { { I }_{ s } }{ { R }_{ g } } \)

It is defined as the deflection produced per unit current flowing through the galvanometer.
I_s  = \(\frac { \theta }{ I } =\frac { NAB }{ K } \Rightarrow I_{ s }=\frac { I }{ G } \).

(i) Obey inverse square law, so they are long-range fields.
(ii) obey the principle of superposition and are linear with respect to source in magnitude,
E ∝ q
B ∝ Idl

If we rotate aright-handed screw by ascrew driver, then the direction of current is same as the direction in which screw advances and the direction of rotation of the screw gives the direction of the magnetic field.

The materials with high initial permeability, low retentivity, low coercivity and thin hysteresis loop with smaller area are preferred to make electromagnets.

The magnitude of the reverse magnetising field for which the residual magnetism of the material vanishes is called its coercivity.

Let the length MN = y and the point P is on its perpendicular bisector. Let O be the point on the conductor as shown in figure. Therefore,
\(OM=ON=\frac { y }{ 2 } ,then\)
\(cos\varphi _{ 1 }=\frac { \frac { y }{ 2 } }{ \sqrt { \frac { { y }^{ 2 } }{ 4 } +{ d }^{ 2 } } } =\frac { adjacent \ length }{ hypotenuse \ length } \)
\(=\frac { ON }{ PH } =-\frac { \frac { y }{ 2 } }{ \sqrt { \frac { { y }^{ 2 } }{ 4 } +{ a }^{ 2 } } } =-\frac { y }{ \sqrt { { y }^{ 2 }+{ 4a }^{ 2 } } } \)
\(cos\varphi _{ 1 }=\frac { adjacent \ length }{ hypotenuse \ length } =\frac { OM }{ PM } \)
\(=-\frac { \frac { y }{ 2 } }{ \sqrt { \frac { { y }^{ 2 } }{ 4 } +{ a }^{ 2 } } } =-\frac { y }{ \sqrt { { y }^{ 2 }+{ 4a }^{ 2 } } } \)
Hence,
\(\overset { \rightarrow }{ B } =\frac { { \mu }_{ ° }I }{ 4\pi a\sqrt { { y }^{ 2 }+{ 4a }^{ 2 } } } \hat { n } \)
For long straight wire, Y\(\rightarrow \infty ,\)
\(\overset { \rightarrow }{ B } =\frac { { \mu }_{ ° }I }{ 2\pi a } \hat { n } \)
The result obtained is the same as we obtained in equation (3.39).

The magnitude of deflecting torque (the torque which makes the object rotate) acting on the bar magnet will tend to align the bar magnet parallel to the direction of the uniform magnetic field \(\overset { \rightarrow }{ B } \) 
\(\left| \overset { \rightarrow }{ r } \right| ={ p }_{ m }Bsin\theta \)
The magnitude of restoring torque acting on the bar magnet can be written as
\(\left| \overset { \rightarrow }{ r } \right| =I\frac { { d }^{ 2 }\theta }{ { dt }^{ 2 } } \)
Under equilibrium conditions, both magnitudes of deflecting torque and restoring torque will be equal but act in the opposite directions, which means
 \(\frac { { d }^{ 2 }\theta }{ { dt }^{ 2 } } =-{ p }_{ m }Bsin\theta \)