Magnetostatics and Biot-Savart’s Law - JEE Topper’s Notes

Key Points:

1. Basic Concepts of Magnetostatics:

• Magnetic Field: –Definition: A region around a current-carrying conductor or a magnet where its influence can be felt is known as a magnetic field. –Representation: It is represented by vectors known as magnetic field lines which indicate the direction and strength of the magnetic field.

2. Biot-Savart Law (NCERT Reference: Class 12 Physics, Chapter 4)

• Mathematical formula to calculate the magnetic field at a point due to a differential current-carrying element.
• The direction of the magnetic field at a point due to a current element is given by the right-hand thumb rule.
• The magnitude of the magnetic field is directly proportional to the current and inversely proportional to the square of the distance from the current element.

3. Magnetic Field due to:

• Straight Current-Carrying Wire: –Applying Biot-Savart’s law, the magnetic field at a distance ‘r’ from a long straight wire carrying current ‘I’ is given by: – $$B=\frac{{\mu }_0}{4\pi }\frac{2I}{r}$$ where (\mu_0) is the permeability of free space.

• Circular Loop: –At the center of a circular loop of radius ‘R’ carrying current ‘I’, the magnetic field is given by: – $$B=\frac{{\mu }_{0}I}{2R}$$ – On the axis of the circular loop, the magnetic field at a distance ‘x’ from the center is given by: –$$B=\frac{{\mu }_{0}I}{4\pi }\left(\frac{R^2}{(R^2+x^2{)}^{3/2}}\right)$$

4. Magnetic Field due to Solenoids (NCERT Reference: Class 12 Physics, Chapter 5)

• A solenoid is a long cylindrical coil of wire closely wound in the form of a helix.
• The magnetic field inside a solenoid is uniform and is given by: –$$B={\mu }_{0}nI$$ where (n) is the number of turns per unit length of the solenoid.

5. Applications of Biot-Savart Law:

• Magnetic Dipole: – A current loop acts like a magnetic dipole, with its north pole pointing in the direction of the magnetic field at its center. – The magnetic dipole moment of a current loop is given by: –$$m=IA$$ where I is the current and A is the area of the loop.
• Torque on Current-Carrying Loop in Magnetic Field: –When a current-carrying loop is placed in a magnetic field, it experiences a torque given by: – $$\overrightarrow{\tau }=\overrightarrow{m}\times \overrightarrow{B}$$ where ( \overrightarrow m ) is the magnetic moment of the loop and ( \overrightarrow B ) is the external magnetic field.

6. Ampere’s law (NCERT Reference: Class 12 Physics, Chapter 4)

• Ampere’s law is a generalization of Biot-Savart’s law and states that the line integral of magnetic field around a closed loop is equal to the total current passing through the surface bounded by the loop.
• $$\oint \overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0\sum I_{enc}$$ where ( \mu_0) is the permeability of free space, ( I_{enc} ) represents the net current flowing through the surface enclosed by the loop.
• Ampere’s law is particularly useful in determining the magnetic field due to symmetric current distributions.

7. Solved Problems and Numerical Practice:

• Practice a wide range of solved examples and numerical problems based on the concepts and equations discussed above to reinforce understanding and hone problem-solving skills.

Reference:

• NCERT Physics Class 11 and 12, CBSE