SATHEE: Notes from Toppers

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

Key Points:

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.

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.

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: – $$B = \frac{\mu_0I}{2R} $$ – On the axis of the circular loop, the magnetic field at a distance ‘x’ from the center is given by: –$$B = \frac{\mu_0I}{4\pi}\left(\frac{R^2}{(R^2+x^2)^{3/2}}\right)$$

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: –$$B = \mu_0nI$$ where (n) is the number of turns per unit length of the solenoid.

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.

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.

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.

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