Magnetostatics- Introduction And Biot Savart Law

Magnetostatics - Introduction and Biot-Savart Law

Learning Objectives

Understand the concept of magnetostatics
Learn about the Biot-Savart Law
Identify the SI unit of magnetic field

1. Introduction to Magnetostatics

Magnetostatics is a branch of electromagnetism
It deals with the study of magnetic fields and their interactions with currents and moving charges
In magnetostatics, the electric fields are considered to be stationary or constant in time

2. Magnetic Field

Magnetic field is a region in space where a magnetic force can be observed
The magnetic field is created by moving electric charges
The strength of the magnetic field is determined by the density of magnetic field lines

3. Magnetic Field Lines

Magnetic field lines represent the direction and strength of the magnetic field
They are always closed curves or loops
The direction of the field lines is from the north pole to the south pole of a magnet

4. Biot-Savart Law - Overview

Biot-Savart Law describes the magnetic field produced by a current-carrying wire
It was formulated by Jean-Baptiste Biot and Félix Savart in the early 19th century
The law states that the magnetic field at a point due to a current element is directly proportional to the current and inversely proportional to the distance from the wire

5. Biot-Savart Law Equation

The Biot-Savart Law can be mathematically represented as: $$ \vec{B} = \frac{{\mu_0}}{{4\pi}} \frac{{I \cdot d\vec{l} \times \vec{r}}}{{r^3}} $$ Where:

$\vec{B}$ is the magnetic field at a point
$I$ is the current passing through the wire
$d\vec{l}$ is a vector element of the wire carrying current
$\vec{r}$ is the vector distance from the element to the point
$r$ is the magnitude of the distance between the element and the point
$\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} , \text{Tm/A}$)

6. Biot-Savart Law - Example 1

Consider a straight conductor carrying a current of 5 A. Determine the magnetic field at a point located 2 meters away from the conductor. Solution:

Given:
- Current (I) = 5 A
- Distance (r) = 2 m
Using the Biot-Savart Law: $$ \vec{B} = \frac{{\mu_0}}{{4\pi}} \frac{{I \cdot d\vec{l} \times \vec{r}}}{{r^3}} $$
Since the conductor is straight, the magnitude of $d\vec{l}$ is equal to the length of the conductor (L)
Therefore, $\vec{B} = \frac{{\mu_0}}{{4\pi}} \frac{{I \cdot L \cdot sin\theta}}{{r^2}}$
The direction of the magnetic field can be determined using the right-hand rule

7. Biot-Savart Law - Example 2

Consider a circular loop of radius 3 cm carrying a current of 2 A. Determine the magnetic field at the center of the loop. Solution:

Given:
- Current (I) = 2 A
- Radius (r) = 0.03 m
Using the Biot-Savart Law: $$ \vec{B} = \frac{{\mu_0}}{{4\pi}} \frac{{I \cdot d\vec{l} \times \vec{r}}}{{r^3}} $$
Since the loop is symmetrical, the magnitude of the magnetic field at the center is constant and can be calculated as: $$ \vec{B} = \frac{{\mu_0 \cdot I \cdot R^2}}{{2 \cdot r^3}} $$
The direction of the magnetic field is perpendicular to the plane of the circular loop

8. Magnetic Field Strength

The strength of a magnetic field can be determined by measuring the force experienced by a moving charge or a current-carrying conductor in the field
The SI unit of magnetic field strength is the Tesla (T)
Magnetic field strength is also commonly expressed in Gauss (G), where 1 T = 10,000 G

9. SI Unit of Magnetic Field

The SI unit of magnetic field is the Tesla (T)
1 Tesla is equal to 1 Newton per Ampere-meter (N/A·m)
The magnetic field is a vector quantity, with both magnitude and direction

10. Summary

Magnetostatics studies the magnetic fields and their interactions with current and moving charges
The Biot-Savart Law describes the magnetic field produced by a current-carrying wire
The SI unit of magnetic field is the Tesla (T)

11. Magnetic Field Due to a Straight Wire

Consider a straight current-carrying wire
- The magnetic field lines around the wire form concentric circles.
- The magnetic field strength decreases with increasing distance from the wire.
The magnitude of the magnetic field at a distance r from the wire can be calculated using the Biot-Savart Law:
- $\vec{B} = \frac{{\mu_0 \cdot I}}{{2\pi \cdot r}}$
- Where I is the current in the wire and $\mu_0$ is the permeability of free space.

12. Magnetic Field Due to a Circular Loop

Consider a circular loop of radius R carrying a current I.
The magnetic field at the center of the loop is given by:
- $\vec{B} = \frac{{\mu_0 \cdot I}}{{2 \cdot R}}$
The magnitude of the magnetic field inside the loop can be calculated using the Biot-Savart Law:
- $\vec{B} = \frac{{\mu_0 \cdot I \cdot r}}{{2 \cdot R^2}}$
The magnetic field outside the loop is negligible.

13. Magnetic Field Due to a Solenoid

A solenoid is a long coil of wire with multiple turns.
Inside the solenoid, the magnetic field is uniform and parallel to the axis of the solenoid.
The magnitude of the magnetic field inside the solenoid is given by:
- $\vec{B} = \mu_0 \cdot n \cdot I$
Where n is the number of turns per unit length and I is the current in the solenoid.

14. Magnetic Field Due to Toroidal Coil

A toroidal coil is a donut-shaped coil of wire.
The magnetic field inside the toroid is strong and uniform.
The magnitude of the magnetic field inside the toroid is given by:
- $\vec{B} = \mu_0 \cdot n \cdot I$
Where n is the number of turns per unit length and I is the current in the toroid.

15. Magnetic Force on a Current-Carrying Conductor

When a current-carrying conductor is placed in a magnetic field, it experiences a magnetic force.
The magnitude of the magnetic force can be calculated using the equation:
- $\vec{F} = I \cdot \vec{L} \times \vec{B}$
Where I is the current, $\vec{L}$ is the length of the conductor, and $\vec{B}$ is the magnetic field.

16. Magnetic Force on a Current-Carrying Wire

The magnetic force on a current-carrying wire can be calculated using the equation:
- $\vec{F} = I \cdot \vec{L} \times \vec{B} \cdot sin\theta$
Where I is the current, $\vec{L}$ is the length of the wire, $\vec{B}$ is the magnetic field, and $\theta$ is the angle between the wire and the magnetic field.

17. Torque on a Current Loop

When a current loop is placed in a magnetic field, it experiences a torque.
The torque can be calculated using the equation:
- $\vec{\tau} = \vec{m} \times \vec{B}$
Where $\vec{m}$ is the magnetic moment of the loop and $\vec{B}$ is the magnetic field.

18. Magnetic Moment of a Current Loop

The magnetic moment of a current loop is a measure of the strength and direction of the loop’s magnetic field.
The magnetic moment of a current loop with N turns and an area A is given by:
- $\vec{m} = N \cdot I \cdot \vec{A}$
Where N is the number of turns, I is the current, and $\vec{A}$ is the area vector.

19. Ampere’s Law

Ampere’s Law is a fundamental law in magnetostatics.
It relates the magnetic field around a closed loop to the current passing through the loop.
The mathematical equation of Ampere’s Law is:
- $\oint \vec{B} \cdot d\vec{s} = \mu_0 \cdot I_{enc}$
Where $\vec{B}$ is the magnetic field, $d\vec{s}$ is the differential element of length along the closed loop, $\mu_0$ is the permeability of free space, and $I_{enc}$ is the current enclosed by the loop.

20. Applications of Magnetostatics

Magnetostatics has numerous practical applications, including:
- Magnetic levitation and Maglev trains
- Magnetic resonance imaging (MRI)
- Electric motors and generators
- Magnetic storage devices (hard drives)
- Particle accelerators (such as cyclotrons)
- Magnetic compasses and navigation systems

21. Magnetic Field and Magnetic Flux

Magnetic Field:
- Magnetic field is a vector quantity that describes the strength and direction of the magnetic force at a given point.
- It is represented by the symbol $\vec{B}$.
- The magnetic field can be created by permanent magnets or by electric currents.
Magnetic Flux:
- Magnetic flux represents the number of magnetic field lines passing through a surface.
- It is denoted by the Greek letter $\Phi$ and is measured in Tesla-square meter (T·m²).
- The magnetic flux through a surface can be calculated using the equation:
  - $\Phi = \vec{B} \cdot \vec{A}$
  - Where $\vec{B}$ is the magnetic field and $\vec{A}$ is the area vector perpendicular to the magnetic field lines.
Examples:
- Magnetic field: Earth’s magnetic field, magnetic field around a bar magnet.
- Magnetic flux: Magnetic flux passing through a closed loop, magnetic flux through a coil.

22. Gauss’s Law for Magnetism

Gauss’s Law for Magnetism is an extension of Gauss’s Law for electric fields.
It relates the magnetic flux through a closed surface to the total magnetic charge enclosed by the surface.
The mathematical equation of Gauss’s Law for Magnetism is:
- $\oint \vec{B} \cdot d\vec{A} = 0$
This implies that magnetic monopoles do not exist, unlike electric charges.

23. Magnetic Induction

Magnetic Induction, also known as Magnetic Field Induction or Magnetic Flux Density, represents the magnetic field created by a current-carrying wire.
It is denoted by the symbol $\vec{B}$.
Magnetic induction can be calculated using the Biot-Savart Law or Ampere’s Law.
Magnetic induction is a vector quantity, meaning it has both magnitude and direction.
Examples:
- Biot-Savart Law: Calculating the magnetic field around a straight wire or a circular loop.
- Ampere’s Law: Determining the magnetic field inside a solenoid or a toroidal coil.

24. Magnetic Field Due to a Current-Loop

When a current-loop is placed in a magnetic field, it interacts with the field and experiences a torque.
The magnitude of the torque can be calculated using the equation:
- $\tau = I \cdot A \cdot \vec{B} \cdot sin\theta$
- Where I is the current, A is the area of the loop, $\vec{B}$ is the magnetic field, and $\theta$ is the angle between the magnetic field and the normal to the plane of the loop.
The magnetic field tries to align the current-loop with its own field direction.

25. Magnetic Field Inside and Outside a Current-Carrying Wire

Inside the wire:
- The magnetic field inside a current-carrying wire can be calculated using Ampere’s Law.
- For an infinitely long straight wire, the magnetic field is directed in concentric circles perpendicular to the wire and its strength decreases as you move further away from the wire.
Outside the wire:
- Outside the wire, the magnetic field becomes weaker and more spread out.
- It follows the right-hand rule where the magnetic field lines form circles around the wire.
Examples:
- Magnetic field inside and outside a current-carrying coil or solenoid.

26. Magnetic Force on a Moving Charge

When a moving charge enters a magnetic field, it experiences a magnetic force.
The magnitude of the magnetic force can be calculated using the equation:
- $\vec{F} = q \cdot \vec{v} \times \vec{B}$
- Where q is the charge of the particle, $\vec{v}$ is its velocity, and $\vec{B}$ is the magnetic field.
The direction of the magnetic force can be determined using the right-hand rule.

27. Cyclotron and Magnetic Deflection

A cyclotron is a type of particle accelerator that uses a combination of electric and magnetic fields to accelerate charged particles.
The particles move in a circular path due to the action of the magnetic field.
The magnetic field provides the necessary centripetal force to keep the particles in a circular path.
By adjusting the magnetic field strength, the trajectory of the particles can be altered.

28. Magnetic Domains

Magnetic domains are regions within a material where the magnetic moments of the atoms or ions are aligned in the same direction.
In the absence of an external magnetic field, the magnetic domains in a material are randomly oriented and cancel each other out.
When a material is placed in an external magnetic field, the magnetic domains align themselves with the field, resulting in the material becoming magnetized.

29. Electromagnetic Induction

Electromagnetic induction occurs when a changing magnetic field induces an electromotive force (emf) or voltage in a circuit.
It is the basis for generating electricity in power plants and the operation of transformers.
The phenomenon was discovered by Michael Faraday and is described by Faraday’s Law.
Faraday’s Law states that the emf induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.

30. Lenz’s Law

Lenz’s Law is a consequence of Faraday’s Law and describes the direction of the induced current in a closed loop.
Lenz’s Law states that the induced current will flow in a direction such that it opposes the change in magnetic flux that caused it.
This law ensures that the conservation of energy is obeyed, as the induced current produces a magnetic field that opposes the change in the applied magnetic field.