Concepts to remember for JEE and CBSE on More Applications of Ampere’s Law:

• Magnetic field at centre and in between two coaxial solenoids

At the center: $$B=\mu_0 (n_1 I_1+n_2 I_2)$$

• In between: $$B= \frac{\mu_0}{4\pi} \frac{2n_1I_1r^2}{[r^2 +(l/2)^2]^{3/2}}$$

• Solenoid: the field inside and its self-inductance

  • Inside the solenoid: $$B=\mu_0nI$$ Self-inductance: $$L=\mu_0n^2A/l$$

• Field due to a toroid or a circular coil in the axial direction and radial direction:

  • On axis: $$B= \frac{\mu_0}{4\pi}\frac{2\pi NI R^2}{x\sqrt{x^2+R^2}}$$ At the centre: $$B= \frac{\mu_0 NI}{2R}$$
  • At radial distance r from the axis: $$B= \frac{\mu_0}{4\pi}\frac{4\pi NI r^2}{\left(R^2+r^2\right)^{3/2}}$$

• Field outside a long straight wire: Straight wire (infinitely long)

$$B= \frac{\mu_0 I}{2\pi r}$$

• Cyclotron or magnetic resonance imaging

  • Cyclotron’s angular frequency: $$ \omega= \frac{qeB}{m}$$
  • Magnetic resonance condition: $$ \omega_0= \gamma B_0$$