### Generalization Of Amperes Law And Its Applications

**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.