Concepts to remember from the “Generalization of Ampere’s law and its applications for JEE and CBSE board exams:**

1. Displacement Current:

• Introduced by James Clerk Maxwell to modify Ampere’s law.
• Represent the time-varying electric field and its effect on the magnetic field.
• Given by: $$I_d = \epsilon_0\frac{\partial \Phi_E}{\partial t}$$, where

• $$I_d$$ is the displacement current
• $$\epsilon_0$$ is the vacuum permittivity
• $$\Phi_E$$ is the changing electric flux.

• Has units of Ampere (A).

2. Maxwell’s Ampere’s Law:

• An extended version of Ampere’s law that includes both conduction current and displacement current.
• Given by: $$\oint \overrightarrow{B} \cdot d\overrightarrow{l}=\mu_0(\sum I_{conduction}+\epsilon_0\frac{d\Phi_E}{dt})$$
• Provides a more complete description of the relationship between electric and magnetic fields.

3. Solenoids and Toroids:

• Solenoid: A long cylindrical coil of closely spaced wire loops that approximates an infinite long wire when current flows through it.
• Toroid: A circular coil of wire forming a donut-like shape, the magnetic field is concentrated entirely inside the toroid.

4. Comparison between Biot-Savart Law and Ampere’s Circuital Law:

• Biot-Savart law provides a direct calculation method for finding the magnetic field due to a current-carrying element.
• Ampere’s Circuital law is useful when dealing with symmetric current distributions, allowing for easier calculations of the magnetic field.
• Both laws are consistent and can be derived from each other.

Applications:

Magnetic field inside a long straight solenoid:

• Uniform magnetic field inside, given by $$B=\mu_0 n I$$, where

• $$n$$ is the number of turns per unit length
• $$I$$ is the current

• Field outside the solenoid is nearly zero.

Magnetic field due to a toroid:

• Uniform magnetic field inside, given by $$B=\frac{\mu_0 N I}{2\pi r}$$, where

• $$N$$ is the total number of turns
• $$I$$ is the current
• $$r$$ is the radius of the toroid

Applications in MRI (Magnetic Resonance Imaging) and Transformers:

• MRI uses strong magnetic fields and changing electric fields to generate images of the human body.
• Transformers use solenoids and toroids to change the voltage of alternating current (AC) electricity.