Magnetostatics- Introduction And Biot Savart Law

Magnetic field due to current carrying loop

Introduction to Magnetostatics
- Study of magnetic fields produced by steady currents
- Importance in various applications such as motors, transformers, and generators
Magnetic Field
- Definition: A region in the space around a magnet or current-carrying wire where a magnetic force can be observed
- Representation: By magnetic field lines
- Direction: North to south outside the magnet or wire
Biot-Savart Law
- Formula: $\mathbf{B} = \frac{{μ_0}}{{4π}} \cdot \frac{{I \cdot \mathbf{dl} \times \mathbf{r}}}{{r^3}}$
  - $\mathbf{B}$: Magnetic field at a point due to current element
  - $μ_0$: Vacuum permeability ($4π \times 10^{-7} , \text{N/A}^2$)
  - $I$: Current flowing through the wire
  - $\mathbf{dl}$: Current element vector
  - $\mathbf{r}$: Position vector from current element to point where magnetic field is to be calculated
  - $r$: Distance between current element and point
Magnetic Field Due to a Current-Carrying Wire
- For an infinitely long straight wire, the magnetic field is given by:
  - $B = \frac{{μ_0 \cdot I}}{{2π \cdot r}}$
  - Direction: Circular path around the wire, as per the right-hand grip rule
Magnetic Field at the Center of a Circular Loop
- For a current-carrying circular loop of radius $R$, the magnetic field at its center is given by:
  - $B = \frac{{μ_0 \cdot I}}{{2R}}$
  - Formula same as that for a straight wire at the center, with $r$ replaced by $R$
Magnetic Field at a Point on the Axis of a Circular Loop
- At a point on the axis of a circular loop, at a distance $x$ from the center, the magnetic field is given by:
  - $B = \frac{{μ_0 \cdot I \cdot R^2}}{{2 \cdot (R^2 + x^2)^{3/2}}}$
Magnetic Field Inside a Solenoid
- A solenoid is a tightly wound helical coil of wire
- For an ideal solenoid, the magnetic field inside the solenoid is uniform and parallel to its axis
- The magnetic field inside the solenoid is given by:
  - $B = μ_0 \cdot n \cdot I$
  - $n$: Number of turns per unit length
Magnetic Field Outside a Solenoid
- The magnetic field outside the solenoid is similar to that of a bar magnet
- For a long solenoid, the magnetic field outside is negligible
Example: The magnetic field at a point on the axis of a circular loop
- Given: Radius of the loop $R = 5 , \text{cm}$, Current through the loop $I = 2 , \text{A}$, Distance $x = 10 , \text{cm}$
- Solution:
  - Substitute the given values in the formula for magnetic field at a point on the axis of a circular loop
  - Calculate the magnetic field at the given point
Example: Magnetic field inside a solenoid
- Given: Number of turns per unit length $n = 1000 , \text{turns/m}$, Current through the solenoid $I = 5 , \text{A}$
- Solution:
  - Substitute the given values in the formula for magnetic field inside a solenoid
  - Calculate the magnetic field inside the solenoid

Ampere’s Law

Ampere’s Law is a mathematical relationship between the magnetic field and the electric current in the circuit.
Formula: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \cdot I$
- $\oint$: Integration along a closed curve
- $\mathbf{B}$: Magnetic field
- $d\mathbf{l}$: Differential length element along the closed curve
- $\mu_0$: Vacuum permeability ($4\pi \times 10^{-7} , \text{N/A}^2$)
- $I$: Total electric current crossing the closed curve

Magnetic Field Due to Straight Infinite Wire Using Ampere’s Law

Using Ampere’s Law, we can also calculate the magnetic field due to a straight infinite wire.
Consider a circular path of radius $r$ around the wire.
Applying Ampere’s Law gives: $B \cdot 2\pi r = \mu_0 \cdot I$
Simplifying, we get: $B = \frac{\mu_0 \cdot I}{2\pi r}$, which is the same as using the Biot-Savart Law.

Magnetic Field Due to Circular Loop Using Ampere’s Law

Using Ampere’s Law, we can calculate the magnetic field at the center of a circular loop.
Consider a circular path of radius $R$ around the loop.
Applying Ampere’s Law gives: $B \cdot 2\pi R = \mu_0 \cdot I$
Simplifying, we get: $B = \frac{\mu_0 \cdot I}{2\pi R}$, which is the same as using the Biot-Savart Law.

Magnetic Field Due to Solenoid Using Ampere’s Law

Using Ampere’s Law, we can determine the magnetic field inside a solenoid.
Consider a path along the length of the solenoid.
Applying Ampere’s Law gives: $B \cdot 2\pi s = \mu_0 \cdot n \cdot I \cdot s$
Simplifying, we get: $B = \mu_0 \cdot n \cdot I$, which is the same as the result obtained using the Biot-Savart Law.

Magnetic Field Due to Toroid Using Ampere’s Law

Using Ampere’s Law, we can find the magnetic field inside a toroid.
Consider a circular path within the toroid with radius $r$.
Applying Ampere’s Law gives: $B \cdot 2\pi r = \mu_0 \cdot n \cdot I \cdot r$
Simplifying, we get: $B = \frac{\mu_0 \cdot n \cdot I}{2\pi r}$, which is the same as the result obtained using the Biot-Savart Law.

Magnetic Field Due to a Helical Coil Using Ampere’s Law

Using Ampere’s Law, we can determine the magnetic field inside a helical coil.
Consider a circular path around the coil with radius $r$.
Applying Ampere’s Law gives: $B \cdot 2\pi r = \mu_0 \cdot n \cdot I \cdot r$
Simplifying, we get: $B = \frac{\mu_0 \cdot n \cdot I}{2\pi r}$, which is the same as the magnetic field due to a long straight wire.

Comparison Between Biot-Savart Law and Ampere’s Law

Biot-Savart Law is used to determine the magnetic field due to a finite length wire or a closed loop.
Ampere’s Law is used to determine the magnetic field for systems with symmetry, such as infinite wires, solenoids, toroids, and helical coils.
Ampere’s Law provides a more convenient and easier method to calculate the magnetic field in such cases.

Magnetic Field Lines

Magnetic field lines are used to visualize the direction and strength of the magnetic field.
Characteristics of magnetic field lines:
- Form closed loops
- Never intersect each other
- Closer the field lines, stronger the magnetic field
- Direction of the field lines indicates the direction of the magnetic field

Magnetic Field Inside a Ferromagnetic Material

Ferromagnetic materials have unique properties that can greatly enhance the magnetic field.
When a ferromagnetic material is placed in an external magnetic field, it becomes magnetized, and the magnetic field inside the material becomes much stronger.
The presence of ferromagnetic materials can greatly affect the behavior of magnetic systems.

Magnetic Field Strength vs. Magnetic Flux Density

Magnetic field strength ($H$) and magnetic flux density ($B$) are related but different concepts.
Magnetic field strength is the measure of the magnetic field created by a current-carrying wire or coil.
Magnetic flux density is the measure of the density of magnetic field lines passing through a given area.
The relationship between $H$ and $B$ is given by $B = \mu_0 \cdot H$, where $\mu_0$ is the vacuum permeability.

Magnetic Force on a Moving Charge

A charge moving in a magnetic field experiences a force perpendicular to both the velocity and the magnetic field.
Formula: $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$
- $\mathbf{F}$: Magnetic force
- $q$: Charge of the particle
- $\mathbf{v}$: Velocity of the particle
- $\mathbf{B}$: Magnetic field

Magnetic Force on a Current-Carrying Conductor

A current-carrying conductor in a magnetic field experiences a force perpendicular to both the current and the magnetic field.
Formula: $\mathbf{F} = \mathbf{I} \times \mathbf{B} \cdot L$
- $\mathbf{F}$: Magnetic force
- $\mathbf{I}$: Current in the conductor
- $\mathbf{B}$: Magnetic field
- $L$: Length of the conductor

Torque on a Current Loop

A current loop in a magnetic field experiences a torque that tends to align the loop with the magnetic field.
Formula: $\mathbf{\tau} = \mathbf{m} \times \mathbf{B}$
- $\mathbf{\tau}$: Torque
- $\mathbf{m}$: Magnetic dipole moment of the loop
- $\mathbf{B}$: Magnetic field

Magnetic Moment of a Current Loop

The magnetic dipole moment of a current loop is a measure of the strength and direction of its magnetic field.
Formula: $\mathbf{m} = \mathbf{I} \cdot \mathbf{A}$
- $\mathbf{m}$: Magnetic dipole moment
- $\mathbf{I}$: Current in the loop
- $\mathbf{A}$: Area of the loop

Magnetic Field Due to a Moving Charge

A moving charge creates a magnetic field around it.
Formula: $\mathbf{B} = \frac{\mu_0}{4\pi} \cdot \frac{q\mathbf{v} \times \mathbf{r}}{r^3}$
- $\mathbf{B}$: Magnetic field at a point due to a moving charge
- $\mu_0$: Vacuum permeability ($4\pi \times 10^{-7} , \text{N/A}^2$)
- $q$: Charge of the moving particle
- $\mathbf{v}$: Velocity of the moving particle
- $\mathbf{r}$: Position vector from the moving charge to the point

Cyclotron Motion

A charged particle moving perpendicular to a magnetic field follows a circular path called cyclotron motion.
The radius of the cyclotron orbit is given by: $r = \frac{mv}{qB}$
- $r$: Radius of the orbit
- $m$: Mass of the particle
- $v$: Velocity of the particle
- $q$: Charge of the particle
- $B$: Magnetic field

Magnetic Field due to a Long Straight Wire using Ampere’s Law

Using Ampere’s Law, we can calculate the magnetic field due to a long straight wire.
Consider a circular path of radius $r$ around the wire.
Applying Ampere’s Law gives: $B \cdot 2\pi r = \mu_0 \cdot I$
Simplifying, we get: $B = \frac{\mu_0 \cdot I}{2\pi r}$, which is the same as using the Biot-Savart Law.

Magnetic Field due to a Circular Loop using Ampere’s Law

Using Ampere’s Law, we can calculate the magnetic field at the center of a circular loop.
Consider a circular path of radius $R$ around the loop.
Applying Ampere’s Law gives: $B \cdot 2\pi R = \mu_0 \cdot I$
Simplifying, we get: $B = \frac{\mu_0 \cdot I}{2\pi R}$, which is the same as using the Biot-Savart Law.

Magnetic Field due to a Solenoid using Ampere’s Law

Using Ampere’s Law, we can determine the magnetic field inside a solenoid.
Consider a path along the length of the solenoid.
Applying Ampere’s Law gives: $B \cdot 2\pi s = \mu_0 \cdot n \cdot I \cdot s$
Simplifying, we get: $B = \mu_0 \cdot n \cdot I$, which is the same as the result obtained using the Biot-Savart Law.

Magnetic Field due to a Toroid using Ampere’s Law

Using Ampere’s Law, we can find the magnetic field inside a toroid.
Consider a circular path within the toroid with radius $r$.
Applying Ampere’s Law gives: $B \cdot 2\pi r = \mu_0 \cdot n \cdot I \cdot r$
Simplifying, we get: $B = \frac{\mu_0 \cdot n \cdot I}{2\pi r}$, which is the same as the result obtained using the Biot-Savart Law.