Moving Charges and Magnetism-01

William Doyle A F

PGT Physics

Lorentz Magnetic Force

A charge q moving through a magnetic field B with a velocity v experiences a force. This is force is given by,

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In scalar form it is,

Where θ is the angle between v and B.

i) If the charge is at rest, i.e. v = 0, then F_m = 0.

So, a stationary charge in a magnetic field does not experience any force.

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ii) If θ = 0° or 180° i.e. if the charge moves parallel or anti-parallel to the direction of the magnetic field, then F_m = 0.

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iii ) If θ = 90° i.e. if the charge moves perpendicular to the magnetic field, then the force is maximum.

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Special Cases:

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(maximum)

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Force on a moving charge in uniform Electric and Magnetic Fields:

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Consider a charge q moves with velocity v in region in which both electric field E and magnetic field B exist. Then it experience both electric and magnetic forces.

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Electric force is,

Magnetic force is,

Net force is,

This is called Lorentz Force.

Lorentz Force is the total force acting on a charge moving in a combined electric and magnetic field.

Force on a current-carrying conductor in a uniform Magnetic Field:

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Force experienced by each electron

Let n = no.density of mobile charges

V_d = drift velocity of charges

Total no. of mobile charge = nlA

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Current I is given by,

Consider a conductor of length l, area of cross section A placed in uniform magnetic field B.

Force on conductor is given by,

Fleming’s Left Hand Rule

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Hold the middle finger, fore finger and thumb of left hand in mutually perpendicular direction such that the middle finger shows direction of current, fore finger shows direction of magnetic field, then thumb shows the direction of force on the current carrying conductor.

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The direction of force on a current carrying conductor in uniform magnetic field is given by Fleming’s Left Hand Rule.

I

B

F

Motion of a charge in Magnetic Field

i) Velocity perpendicular to Magnetic Field

Magnetic Force

Since,

Centripetal force ,

Centripetal force is provided by the magnetic force,

Time period is given by,

Frequency is given by,

Angular Frequency is given by,

This is also called Cyclotron frequency.

ii) Velocity at an angle to Magnetic Field

X

Y

v

B

Due to the v_⊥ the charge will execute circular motion perpendicular to magnetic field.

The distance moved along the magnetic field in one time period is called pitch p.

Due to v|| the charge will have a transverse motion along the direction of magnetic field in addition to circular motion. Due to the combined effect of circula and transverse motion the charge will move along helical path.

Pitch(p)

Pitch

Velocity Selector

Consider case in which electric and magnetic fields are perpendicular to each other and also perpendicular to the velocity of the particle.

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When electric and magnetic forces are in opposite directions the total force on the charge is zero and the charge will move in the fields undeflected.

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Y

Z

X

E

F_e

v

F_m

B

Cyclotron

The cyclotron is a machine to accelerate charged particles or ions to high energies.

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Principle

A charged particle made to repeatedly move along a circular path of increasing radius in a crossed electric and magnetic field attains high energy.

Cyclotron

Working: Imagining D₁ is positive and D₂ is negative, the + vely charged particle kept at the centre and in the gap between the dees get accelerated towards D₂. Due to perpendicular magnetic field and according to Fleming’s Left Hand Rule the charge gets deflected and describes semi-circular path.

When it is about to leave D₂, D₂ becomes + ve and D₁ becomes – ve. Therefore the particle is again accelerated into D₁ where it continues to describe the semi-circular path. The process continues till the charge traverses through the whole space in the dees and finally it comes out with very high speed through the window.

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Theory

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The magnetic force experienced by the charge provides centripetal force required to describe circular path.

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If T is the time period of the high frequency oscillator, then for resonance,

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The frequency of circular motion is

Frequency is is independent of the radius of the circular path.

Maximum Energy of the Particle:

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Maximum Kinetic Energy of the charged particle is when r = R (radius of the Dees).

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