# Magnetic Force and Motion in a Magnetic Field

• When current is flowing through a conductor only magnetic field is produced around the conductor which is non conservative.
• If charge +q is moving with velocity \tt \overline{V}, making an angle ‘θ’ with the direction of field, force acting on the charge is \tt \overline{F} = q. \left(\overline{V} \times \overline{B}\right) = qVB \sin \theta
• The radius of the circular path when a positive charge is projected perpendicular to the magnetic field is given by \tt r = \frac{mv}{Bq} = \frac{P}{Bq} =\sqrt{ \frac{2m KE}{Bq}} (Since KE = kinetic energy)
• The time period of rotation \tt T = \frac{2 \pi r}{V} = \frac{2 \pi m}{Bq}
• When the particle enters the magnetic field at an angle ‘θ’ with B (θ ≠ 0°  θ ≠ 90° and θ ≠ 180°) Then the path followed by the particle will be helical and radius of helix = \tt \frac{mv \sin \theta}{Bq}
• Distance travelled by the particle along magnetic field in one complete rotation or pitch of the helix is \tt P = V \cos \theta T = \frac{2 \pi m V \cos \theta}{Bq}
• When a charged particle enters a region of both electric and magnetic fields the force is called Lorentz force F = Fe + Fm = \tt q [E + (\overline{V} \times \overline{B}]
• Force acting on a current carrying conductor in a magnetic field \tt F = i \left(\overline{l} \times \overline{B}\right) = B i l \sin \theta
• If the conductor is semi-circular then force is F = Bi(2r) sin θ

### View the Topic in this video From 0:31 To 10;06

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1. Force on a charged particle in a uniform magnetic field
\overrightarrow{F} = q(\overrightarrow{v} \times \overrightarrow{B}) \tt\ or\ F = qvB \sin \theta

2. Radius of circular path is R = \frac{mv}{Bq} = \frac{\sqrt{2 mK}}{qB}

3. Time period of revolution is T = \frac{2 \pi R}{v} = \frac{2 \pi m}{qB}

4. The frequency, v = \frac{1}{T} = \frac{qB}{2 \pi m}

5. The force experienced by a straight conductor of length l carrying current I when placed in a uniform magnetic field \overrightarrow{B} is \overrightarrow{F} = I(\overrightarrow{l} \times \overrightarrow{B}); F = IlB sin θ