More Applications of Ampere’s Law - JEE Topper’s Detailed Notes

Solenoids:

• Magnetic field inside a solenoid: $$ B = \mu_0 n I$$ Where,

• (\mu_0) is the permeability of free space

• (n) is the number of turns per unit length

• (I) is the current flowing through the solenoid

• Magnetic moment of a solenoid: $$ M = nIA $$ Where, (A) is the cross-sectional area of the solenoid.

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Toroids:

• Magnetic field inside a toroid: $$B = \frac{\mu_0 N I}{2\pi r}$$ Where,
• (N) is the number of turns
• (I) is the current
• (r) is the radius of the toroid.

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Magnetic Field of a Current-Carrying Wire:

• Magnetic field due to a long straight wire: $$ B = \frac{\mu_0}{4\pi} \frac{2I}{d}$$ Where,
• (\mu_0) is the permeability of free space
• (I) is the current
• (d) is the perpendicular distance from the wire.

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Applications in Electromagnetism:

Electromagnets

• Solenoids can be used to create powerful electromagnets by wrapping a large number of turns of wire around a ferromagnetic core.
• Ferromagnetic materials such as iron and nickel become strongly magnetized when exposed to a magnetic field, greatly enhancing the magnetic field strength.

Transformers

• Transformers utilize Ampere’s law to transfer electrical energy from one circuit to another through electromagnetic induction.
• By varying the number of turns in the primary and secondary coils, transformers can alter the voltage and current levels while maintaining power balance.

Motors

• Electric motors convert electrical energy into mechanical energy based on Ampere’s law and the interaction between magnetic fields and current-carrying conductors.
• When current flows through a coil placed in a magnetic field, it experiences a magnetic force that causes it to rotate.

Reference: NCERT Class 12, Chapter 7 (Alternating Current)

Biot-Savart Law:

The Biot-Savart law provides a general formula to calculate the magnetic field at a point due to a current element:

$$ d\overrightarrow{B} = \frac{\mu_0}{4\pi} \frac{Id\overrightarrow{l} \times \hat{r}}{r^2} $$

Where:

• (d\overrightarrow{l}) is the vector representing the length and orientation of the current element
• (\hat{r}) is the unit vector pointing from the current element to the observation point
• (r) is the distance from the current element to the observation point

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Magnetic Force between Two Parallel Currents:

The magnetic force between two parallel current-carrying wires can be calculated using Ampere’s law:

$$F = \frac{\mu_0 I_1 I_2 \ell}{2\pi d} $$

Where:

• (F) is the magnetic force
• (I_1) and (I_2) are the currents in the two wires
• (\ell) is the length of the wires
• (d) is the distance between the wires.

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Amperian Loops and Symmetry:

Amperian loops are imaginary paths used to apply Ampere’s law in situations with symmetry, simplifying the calculation of magnetic fields. By exploiting symmetry, appropriate Amperian loops can be chosen to reduce complex geometries into simpler ones.

Reference: NCERT Class 12, Chapter 6 (Magnetic Fields)

Applications in Magnetic Resonance Imaging (MRI):

Magnetic Resonance Imaging (MRI) utilizes strong magnetic fields generated using Ampere’s law.

• Strong magnetic fields align the spins of protons in the human body.
• Radiofrequency waves are then applied, causing the spins to flip.
• As the spins relax, they emit radio waves that are detected and processed to generate detailed images of the human body.

Reference: Beyond NCERT, consult specialized texts or online resources for medical physics or MRI technology.