Moving Charges and Magnetism-Revision Notes

                                                         CBSE Class 12 Physics

Revision Notes
Chapter-4
Moving Charges and Magnetism

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• Biot – Savart law: The magnitude of Magnetic Field dB→$d\overrightarrow{B}$ is proportional to the steady current I due to an element dl→$d\overrightarrow{l}$ at a point P and inversely proportional to the distance r from the current element is, dB→=μ04πIdl→×r→r3$d\overrightarrow{B}=\frac{{\mu }_0}{4\pi }I\frac{d\overrightarrow{l}\times \overrightarrow{r}}{r^3}$
• Magnetic field due to long straight current carrying conductor:
  B=μoI4πa[sinϕ1+sinϕ2]$B=\frac{{\mu }_{o}I}{4\pi a}\left[\sin {\varphi }_1+\sin {\varphi }_2\right]$
• If conductor is infinitely long, B=μoI2πa$B=\frac{{\mu }_{o}I}{2\pi a}$
• Right-hand rule is used to find the direction of magnetic field due to straight current carrying conductor.
• Force on a Straight Conductor: Force F on a straight conductor of length l$l$ and carrying a steady current I placed in a uniform external magnetic field B, F→=I(l→×B→)$\overrightarrow{F}=I(\overrightarrow{l}\times \overrightarrow{B})$
• Lorentz Force: Force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E is given by F→=q(v→×B→+E→)$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B}+\overrightarrow{E})$
• Magnetic Force: The magnetic force FB−→=q(v→×B→)$\overrightarrow{F_B}=q(\overrightarrow{v}\times \overrightarrow{B})$ is normal to v→$\overrightarrow{v}$ and work done by it, is zero.
• Cyclotron: A charge q executes a circular orbit in a plane normal to B with frequency called the cyclotron frequency given by, vc=qB2πm$v_c=\frac{qB}{2\pi m}$
  This cyclotron frequency is independent of the particle’s speed and radius.
• Magnetic Field due to Circular current carrying Coil: Magnetic field due to circular coil of radius 'a' carrying a current I at an axial distance r from the centre-
  B=μ0Ia22(r2+a2)3/2$B=\frac{{\mu }_{0}I{a}^2}{2{(r^2+a^2)}^{3/2}}$
  At the centre of the coil, B=μ0I2a$B=\frac{{\mu }_{0}I}{2a}$
• Ampere’s Circuital Law: For an open surface S bounded by a loop C, then the Ampere’s law states that
  ∮CB→.dl→=μ0I$\underset{C}{\oint }\overrightarrow{B}.d\overrightarrow{l}={\mu }_{0}I$
  where I refers to the current passing through S. If B is directed along the tangent to every point on the perimeter, then
  BL=μ0Ie$BL={\mu }_0{I}_e$
  where Ie is the net current enclosed by the closed circuit.
• Magnetic field at a distance R from a long, straight wire carrying a current I is given by, B=μ0I2πR$B=\frac{{\mu }_{0}I}{2\pi R}$
  The field lines are circles concentric with the wire.
• Magnetic field B inside a long Solenoid carrying a current I is 
  B=μ0nI$B={\mu }_{0}nI$
  where n is the number of turns per unit length.
• For a toroid, B=μ0NI2πr$B=\frac{{\mu }_{0}NI}{2\pi r}$ where N is the total number of turns and r is the average radius.
• Magnetic Moment of a Planar Loop: Magnetic moment m of a planar loop carrying a current I, having N closely wound turns, and an area A, is m→=NIA→$\overrightarrow{m}=NI\overrightarrow{A}$
  and direction of m→$\overrightarrow{m}$ is given by the Right – Hand Thumb Rule: Curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of m→$\overrightarrow{m}$ (and A→$\overrightarrow{A}$).
• When this loop is placed in a uniform magnetic field B, then, the force F on it is, F = 0 and the torque on it is, τ→=m→XB→$\overrightarrow{\tau }=\overrightarrow{m}X\overrightarrow{B}$
  In a moving coil galvanometer, this torque is balanced by a counter torque due to a spring, yielding
  kϕ=NI AB$k\varphi =NI\text{ AB}$ where ϕ$\varphi$ is the equilibrium deflection and k is the torsion constant of the spring.
• Magnetic Moment in an Electron: An electron moving around the central nucleus has a magnetic moment μl${\mu }_l$, given by
  μl=e2ml${\mu }_l=\frac{e}{2m}l$
  where l$l$ is the magnitude of the angular momentum of circulating electron about the central nucleus.
• The smallest value of μl${\mu }_l$ is called the Bohr magneton μB and it is 
  μB = 9.27 ×$\times$ 10-24 J/T.