Concepts to remember for JEE and CBSE board exams on Magnetostatics- Introduction And Biot Savart Law

1. **Magnetic Field of a Current-Carrying Wire:** - **Right-Hand Rule for Current-Carrying Wire:** Grasp the wire with the right hand, with the thumb pointing in the direction of the conventional current. The fingers wrap around the wire, and the direction they curl indicates the direction of the magnetic field. - **Magnetic Field Strength Due to a Single Current-Carrying Wire:** $$B = \frac{\mu_0 \times I}{4\pi r}$$ where \(B\) is the magnetic field strength, \(\mu_0\) is the permeability of free space (\(4\pi\times10^{-7} \ \text{Tm/A}\)), \(I\) is the current, and \(r\) is the distance from the wire.
2. **Biot-Savart Law:** - **Mathematical Expression of Biot-Savart Law:** $$d\overrightarrow{B}=\frac{\mu_0}{4\pi}\frac{I\overrightarrow{dl}\times\hat{r}}{r^2}$$ where \(\overrightarrow{B}\) is the differential magnetic field vector, \(I\) is the current, \(d\overrightarrow{l}\) is a differential length vector of the current-carrying wire, \(\mu_0\) is the permeability of free space, \(\hat{r}\) is a unit vector from the current element to the observation point, and \(r\) is the distance between the current element and the observation point. - **Direction of Magnetic Field Due to Biot-Savart Law:** The direction of the magnetic field is given by the cross product of the current \(\overrightarrow{I}\) and the displacement \(\overrightarrow{l}\) vectors and follows the right-hand rule.
3. **Magnetic Field of Simple Geometrical Shapes:** - **Magnetic Field at the Center of a Circular Current Loop:** $$B = \frac{\mu_0 I}{2R}$$ where \(B\) is the magnetic field strength at the center, \(\mu_0\) is the permeability of free space, \(I\) is the current, and \(R\) is the radius of the circular loop. - **Magnetic Field on the Axis of a Solenoid:** $$B = \mu_0 nI$$ where \(B\) is the magnetic field strength, \(n\) is the number of turns per unit length of the solenoid, \(I\) is the current, and \(\mu_0\) is the permeability of free space. - **Magnetic Field of a Toroid:** $$B = \frac{\mu_0 N I}{2\pi r}$$ where \(B\) is the magnetic field strength within the toroid, \(\mu_0\) is the permeability of free space, \(N\) is the number of turns, \(I\) is the current, and \(r\) is the radius of the toroid.
4. **Applications of Biot-Savart Law:** - **Calculating Magnetic Field of a Current-Carrying Coil:** Divide the coil into small current elements, determine their differential magnetic field contributions, and integrate to find the total magnetic field. - **Calculating Force between Two Current-Carrying Wires:** Use Biot-Savart Law to calculate the magnetic field at the location of one wire due to the other wire and then apply the Lorentz force equation to determine the force.