Electromagnetism - Magnetic Field

Magnetic Field

If a charged particle having charge q mass m enters a magnetic field B with a velocity v then the force experienced by the particle is

F = q (v x B)

This principle is used in television receiver for deflection of electrons. The unit of magnetic field is 

B = Wb/m ^-2 or teals

The CGS unit is gauss = Maxwell/cm²

If electric and magnetic fields are present then the force experienced is called Lorentz force and is given by 

F = q E + q (v x B₀)

J.J. Thomson used this method to fined e/m

If the velocity of a charged particle is always perpendicular to the magnetic field then it describes a circle of radius R such that

R = mv/qB

Time period of revolution

T = 2π/v = 2πm/qB

Cyclotron frequency 

F = I/T = qB/2 πm

If the charged particle enters obliquely in a magnetic field then the velocity can be resolved into x and y components the y perpendicular component describes a circle and x parallel component describes linear motion. Therefore, the particle moves in a helix. 

The horizontal or linear distance moved in one complete revolution is called the pitch of the helix.

Pitch of the helix = vxT = vx (2πm/qB)

Magnetic force of due to a current carrying conductor

d F = I dI x B 

the direction of force is determined using Fleming's left hand rule. 

If the conductor is straight F = 1 I x B

Otherwise integrate to find F = ∫1 dI x B

Torque acting on a current carrying loop τ = I A x B

If there are n loops τ  = nI A x B where A is area vector A = Ib

τ = M x B' where M = IA or nIA is magnetic dipole moment. 

Biot savart law 

d_B = μ₀/4π I dI x r/r³

is the magnetic field due to a length dl of a conductor carrying current I at a distance r from the conductor.

dB = π₀ IdI din ∅ / 4 π² where μ₀ = 4π x 10 ^{- 7} Wb (A-m) ^-1 or henerym^-1 


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