Behavior of a charge in Magnetic field

Charge feels force on it. The force is dependent on the charge (q), velocity(v)and magnetic field (B).

$$F=q(v \times B)$$

$$\mathrm{F}=\mathrm{qvB} \sin \theta$$

Let us consider this particle has a charge q and it moves in the direction of magnetic field B (motion in a magnetic field); the velocity is v and θ is the angle between B and V

We have different cases for a force experienced by a charge

CASE 1: if θ = 0 or 180

Velocity of charge is parallel or anti-parallel to magnetic field.

F= 0.

Trajectory = straight line.

CASE 2: if θ = 90

Velocity of charge is perpendicular to magnetic field

$$F=q v B \sin 90$$

$$F=qvB$$

F = F centripetal

Trajectory = circle (right-hand thumb rule )

Biot savart law

Biot savart law is a fundamental relationship between an electric current I and the magnetic field B in physics.

Biot savart law equations describe the relationship between the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and electric current.

$$d B=\frac{\mu_0 I d l \sin \theta}{4 \pi r^2}$$

dL = infinitesimal length of a conductor carrying electric current.

Magnetic Field

Magnetic Field is a region of space near a magnet.

In other words

A magnetic field is a field that describes the magnetic influence on moving electric charges and electric currents. A moving charge(q) in a magnetic field experiences a force perpendicular to its own velocity (v) and to the magnetic field. It can be denoted with B.

The electric field produced by a point charge q at rest at the origin is

$$E=F / q$$

Where F is Electrostatics Force, and q is point charge.

Magnetic field lines

The magnetic field lines are a visual and intuitive realisation of the magnetic field.

The dipole in a uniform magnetic field

The magnetic field lines give us an

approximate idea of the magnetic field (B). To

determine the magnitude of B accurately. This is done by placing a small compass needle of known magnetic moment (m) and moment of inertia (I)

and allowing it to oscillate in the magnetic field.

The torque on the needle is

$$\tau=m \times B$$
In magnitude

$$\tau=m B \sin \theta$$

Here τ is restoring torque and θ is the angle between m and B.