Magnetostatics- Introduction And Biot Savart Law

Magnetostatics - Introduction And Biot-Savart Law - Magnetic Field Values

Introduction to Magnetostatics
- Magnetostatics is the study of magnetic fields produced by static or constant currents.
- It deals with the behavior of permanent magnets and the interaction of current-carrying conductors.
- Unlike electrostatics, magnetostatics involves moving charges and currents.
Biot-Savart Law
- Biot-Savart Law is used to determine the magnetic field produced by a current-carrying conductor.
- Mathematically, it is represented as: $$\mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{{I \mathbf{dl} \times \mathbf{r}}}{r^3}$$
- Where:
  - $\mathbf{B}$ is the magnetic field vector at a point.
  - $\mu_0$ is the permeability of free space.
  - $I$ is the current flowing through the conductor.
  - $\mathbf{dl}$ is an element of the conductor carrying the current.
  - $\mathbf{r}$ is the position vector from the element $\mathbf{dl}$ to the point where magnetic field is to be determined.
  - $r$ is the magnitude of vector $\mathbf{r}$.
Magnetic Field Values
- Magnetic field depends on the current through the conductor, distance from the conductor, and the shape of the conductor.
- The direction of the magnetic field is given by the right-hand rule.
- Magnetic field lines form closed loops around the current-carrying conductor.
- Inside the conductor, magnetic field lines form concentric circles.
Magnetic Field Due to a Straight Conductor
- For an infinitely long straight wire carrying current $I$:
  - The magnetic field at a point P at a perpendicular distance $r$ from the wire is given by:
  - $$B = \frac{\mu_0 I}{2\pi r}$$
  - The magnetic field lines are in the form of concentric circles centered around the wire.
Magnetic Field Due to a Circular Loop
- For a circular loop of radius $R$ carrying current $I$:
  - The magnetic field at the center (along the axis of the loop) is given by:
  - $$B = \frac{\mu_0 I}{2R}$$
  - The magnetic field lines are symmetrical about the axis of the loop.
Magnetic Field Inside a Solenoid
- A solenoid is a coil of wire wound into a helical shape.
- Inside a long solenoid carrying current $I$:
  - The magnetic field is uniform and parallel to the axis of the solenoid.
  - The magnitude of the magnetic field is given by:
  - $$B = \mu_0 n I$$
  - Where $n$ is the number of turns per unit length of the solenoid.
Magnetic Field Around a Current-Carrying Coil
- For a current-carrying coil, the magnetic field at a point along the axis of the coil is given by:
- $$B = \frac{\mu_0 n I}{2R}$$
- Where $n$ is the number of turns per unit length and $R$ is the radius of the coil.
Magnetic Field Due to a Straight Current-Carrying Conductor Using Ampere’s Law
- Ampere’s Law can be used to determine the magnetic field around a straight current-carrying conductor.
- The magnetic field can be calculated using the formula:
- $$B = \frac{\mu_0 I}{2\pi r}$$
- Where $I$ is the total current passing through a surface bounded by the closed loop.
Magnetic Field Inside a Toroid
- A toroid is a solenoid wound into a donut-like shape.
- Inside a toroid carrying current $I$:
  - The magnetic field is uniform and parallel to the axis of the toroid.
  - The magnitude of the magnetic field is given by:
  - $$B = \mu_0 n I$$
  - Where $n$ is the number of turns per unit length and $I$ is the current.
Force Between Two Parallel Current-Carrying Conductors
- When two long parallel conductors carry currents $I_1$ and $I_2$, the force between them can be calculated using:
- $$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$
- Where $L$ is the length of the conductors and $d$ is the distance between them.

Magnetic Field Due to a Circular Loop - Example

Consider a circular loop of radius $R$ carrying a current $I$.
Using the formula for the magnetic field at the center of the loop: $B = \frac{\mu_0 I}{2R}$
Let’s calculate the magnetic field for a loop with $R = 0.1$ m and $I = 2$ A.
Substituting the values: $B = \frac{(4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(2 , \text{A})}{2(0.1 , \text{m})}$
Solving the equation, we find: $B = 1.27 \times 10^{-5}$ T

Magnetic Field Inside a Solenoid - Example

Consider a solenoid with $n = 200 , \text{turns/m}$ and carrying a current $I = 0.5 , \text{A}$.
Using the formula for the magnetic field inside a solenoid: $B = \mu_0 n I$
Let’s calculate the magnetic field for this solenoid.
Substituting the values: $B = (4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(200 , \text{turns/m})(0.5 , \text{A})$
Solving the equation, we find: $B = 1.26 \times 10^{-4}$ T

Magnetic Field Around a Current-Carrying Coil - Example

Consider a coil with $n = 500 , \text{turns/m}$ and a radius $R = 0.2 , \text{m}$ carrying a current $I = 0.8 , \text{A}$.
Using the formula for the magnetic field around a current-carrying coil: $B = \frac{\mu_0 n I}{2R}$
Let’s calculate the magnetic field at a point on the axis of the coil.
Substituting the values: $B = \frac{(4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(500 , \text{turns/m})(0.8 , \text{A})}{2(0.2 , \text{m})}$
Solving the equation, we find: $B = 2.05 \times 10^{-3}$ T

Magnetic Field Due to a Straight Current-Carrying Conductor Using Ampere’s Law - Example

Let’s consider a straight wire carrying a current $I = 4 , \text{A}$ and a distance $d = 0.15 , \text{m}$ from the wire.
We want to calculate the magnetic field at this distance using Ampere’s Law.
Using the formula: $B = \frac{\mu_0 I}{2\pi r}$
Substituting the values: $B = \frac{(4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(4 , \text{A})}{2\pi(0.15 , \text{m})}$
Solving the equation, we find: $B = 4.25 \times 10^{-6}$ T

Magnetic Field Inside a Toroid - Example

Consider a toroid with a current $I = 1.5 , \text{A}$ and $n = 300 , \text{turns/m}$.
Using the formula for the magnetic field inside a toroid: $B = \mu_0 n I$
Let’s calculate the magnetic field for this toroid.
Substituting the values: $B = (4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(300 , \text{turns/m})(1.5 , \text{A})$
Solving the equation, we find: $B = 5.65 \times 10^{-4}$ T

Force Between Two Parallel Current-Carrying Conductors - Example

Let’s consider two parallel conductors carrying currents $I_1 = 2 , \text{A}$ and $I_2 = 3 , \text{A}$, separated by a distance $d = 0.1 , \text{m}$.
We want to calculate the force between them using the formula: $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$
Let’s assume the length of the conductors $L = 0.5 , \text{m}$.
Substituting the values: $F = \frac{(4\pi \times 10^{-7} , \text{T} \cdot \text{m/A})(2 , \text{A})(3 , \text{A})(0.5 , \text{m})}{2\pi(0.1 , \text{m})}$
Solving the equation, we find: $F = 1.2 \times 10^{-4}$ N

Magnetic Field Due to a Current Loop - Calculation

Consider a circular current loop of radius $R$ carrying a current $I$.
To calculate the magnetic field at a point on the axis of the loop, we can use the equation:
$$B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}$$
Where $z$ is the distance between the point and the center of the loop along the axis.
This equation is derived from the Biot-Savart Law.
It is valid for points on the axis far away from the loop, where $z » R$.

Magnetic Flux - Definition and Calculation

Magnetic Flux is a measure of the amount of magnetic field passing through a surface.
It is defined as the product of the magnetic field $\mathbf{B}$ and the area vector $\mathbf{A}$: $$\Phi = \mathbf{B} \cdot \mathbf{A}$$
The unit of magnetic flux is Weber (Wb) or Tesla-meter squared (T·m^2).
The magnetic flux through a closed surface is always zero (unless magnetic monopoles exist).
For a uniform magnetic field, the magnetic flux through a surface is given by: $$\Phi = B \cdot A \cdot \cos(\theta)$$
Where $B$ is the magnitude of the magnetic field, $A$ is the area of the surface, and $\theta$ is the angle between $\mathbf{B}$ and $\mathbf{A}$.

Gauss’s Law for Magnetism - Introduction

Gauss’s Law for Magnetism is an important law in magnetism similar to Gauss’s Law for Electric Fields.
It states that the magnetic flux through any closed surface is always zero, i.e., $\Phi = 0$.
Gauss’s Law for Magnetism implies that there are no magnetic monopoles (single magnetic charges).
It is a fundamental law in magnetostatics and is based on experimental observations.
Gauss’s Law for Magnetism is mathematically equivalent to the statement that the magnetic field is divergence-free, i.e., $\nabla \cdot \mathbf{B} = 0$.
The absence of magnetic monopoles ensures that magnetic field lines are always closed loops.

Ampere’s Law - Introduction

Ampere’s Law is another fundamental law in magnetism.
It relates the magnetic field around a closed loop to the current passing through the loop.
Mathematically, Ampere’s Law is represented as: $$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$$
Where $\mathbf{B}$ is the magnetic field, $d\mathbf{l}$ is an element of the loop, $\mu_0$ is the permeability of free space, and $I_{\text{enclosed}}$ is the total current enclosed by the loop.
Ampere’s Law is similar to Gauss’s Law for Electric Fields, but it applies to magnetic fields and closed loops.
It is a useful tool for calculating magnetic fields, especially in situations where symmetry exists.

Application of Ampere’s Law - Solenoid

A solenoid is a tightly wound helical coil of wire, often with an iron core.
The magnetic field inside a solenoid is nearly uniform and strong.
The effect of a solenoid is similar to that of a bar magnet.
Ampere’s Law can be applied to calculate the magnetic field inside a solenoid.
The equation for the magnetic field inside a solenoid is:
- $$B = \mu_0 n I$$
Where $B$ is the magnetic field, $\mu_0$ is the permeability of free space, $n$ is the number of turns per unit length, and $I$ is the current through the solenoid.

Magnetic Field Due to a Current Loop - Calculation

Consider a circular current loop of radius $R$ carrying a current $I$.
To calculate the magnetic field at a point on the axis of the loop, we can use the equation:
- $$B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}$$
Where $z$ is the distance between the point and the center of the loop along the axis.
This equation is derived from the Biot-Savart Law.
It is valid for points on the axis far away from the loop, where $z » R$.
The magnetic field decreases as the distance from the loop increases.

Magnetic Flux - Definition and Calculation

Magnetic Flux is a measure of the amount of magnetic field passing through a surface.
It is defined as the product of the magnetic field $\mathbf{B}$ and the area vector $\mathbf{A}$:
- $$\Phi = \mathbf{B} \cdot \mathbf{A}$$
The unit of magnetic flux is Weber (Wb) or Tesla-meter squared (T·m^2).
The magnetic flux through a closed surface is always zero (unless magnetic monopoles exist).
For a uniform magnetic field, the magnetic flux through a surface is given by:
- $$\Phi = B \cdot A \cdot \cos(\theta)$$
Where $B$ is the magnitude of the magnetic field, $A$ is the area of the surface, and $\theta$ is the angle between $\mathbf{B}$ and $\mathbf{A}$.

Gauss’s Law for Magnetism - Introduction

Gauss’s Law for Magnetism is an important law in magnetism similar to Gauss’s Law for Electric Fields.
It states that the magnetic flux through any closed surface is always zero, i.e., $\Phi = 0$.
Gauss’s Law for Magnetism implies that there are no magnetic monopoles (single magnetic charges).
It is a fundamental law in magnetostatics and is based on experimental observations.
Gauss’s Law for Magnetism is mathematically equivalent to the statement that the magnetic field is divergence-free, i.e., $\nabla \cdot \mathbf{B} = 0$.
The absence of magnetic monopoles ensures that magnetic field lines are always closed loops.

Ampere’s Law - Introduction

Ampere’s Law is another fundamental law in magnetism.
It relates the magnetic field around a closed loop to the current passing through the loop.
Mathematically, Ampere’s Law is represented as:
- $$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$$
Where $\mathbf{B}$ is the magnetic field, $d\mathbf{l}$ is an element of the loop, $\mu_0$ is the permeability of free space, and $I_{\text{enclosed}}$ is the total current enclosed by the loop.
Ampere’s Law is similar to Gauss’s Law for Electric Fields, but it applies to magnetic fields and closed loops.
It is a useful tool for calculating magnetic fields, especially in situations where symmetry exists.

Magnetic Field Due to a Straight Current-Carrying Conductor

For an infinitely long straight wire carrying current $I$:
- The magnetic field at a point P at a perpendicular distance $r$ from the wire is given by:
  - $$B = \frac{\mu_0 I}{2\pi r}$$
- The magnetic field lines are in the form of concentric circles centered around the wire.
The magnitude of the magnetic field decreases with increasing distance from the wire.
The direction of the magnetic field is given by the right-hand rule, using the direction of the current and the curling of fingers.

Magnetic Field Due to a Circular Loop

For a circular loop of radius $R$ carrying current $I$:
- The magnetic field at the center (along the axis of the loop) is given by:
  - $$B = \frac{\mu_0 I}{2R}$$
- The magnetic field lines are symmetrical about the axis of the loop.
The magnetic field at the center of the loop is stronger than at points closer to the edges of the loop.
The direction of the magnetic field is perpendicular to the plane of the