U3AOS2 Topic 6: Magnetic Fields

Introduction
Magnetic fields are a fundamental aspect of electromagnetism, one of the four fundamental forces of nature. They are essential in various physical phenomena and technologies, from guiding charged particles in accelerators to operating electric motors and generating electricity in power plants. This comprehensive exploration covers the nature of magnetic fields, how they are generated, their properties, and their applications.
Definition and Nature of Magnetic Fields
A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion. It is represented by the symbol $\mathbf{B}$ and is measured in teslas (T) in the International System of Units (SI). The field exerts forces on moving charges and magnetic materials, and its effects can be observed through the interaction with other magnetic fields and currents.
Origin of Magnetic Fields
Magnetic fields are generated by:
Moving Electric Charges:
The most fundamental source of a magnetic field is a moving electric charge. According to Ampère's Law, an electric current (which is a flow of electric charges) generates a magnetic field around it. The direction of this magnetic field can be determined using the right-hand rule: if you point the thumb of your right hand in the direction of the current, the curled fingers show the direction of the magnetic field lines.
Magnetic Materials:
Certain materials, such as iron, nickel, and cobalt, exhibit intrinsic magnetic properties due to the alignment of their atomic magnetic moments. In these materials, the magnetic field is generated by the alignment of the magnetic dipoles at the atomic level.
Earth’s Magnetic Field:
The Earth itself generates a magnetic field due to the movement of molten iron and other metals in its outer core. This geodynamo effect creates a large-scale magnetic field that extends into space.
Properties of Magnetic Fields
Magnetic Field Lines:
Magnetic fields are visualized using field lines that represent the direction and strength of the field. These lines emerge from the north pole of a magnet and enter the south pole. The density of the lines indicates the strength of the magnetic field: the closer the lines, the stronger the field.
Direction and Magnitude:
The direction of the magnetic field at a point is defined as the direction of the force it exerts on a positive moving charge. The magnitude of the field is proportional to the force experienced by a moving charge. The magnetic field $\mathbf{B}$ at a point in space can be expressed in terms of its components in a coordinate system.
Magnetic Flux:
Magnetic flux $\Phi$ is a measure of the total magnetic field passing through a given area. It is calculated by the product of the magnetic field $\mathbf{B}$ and the area $A$ through which the field lines pass, and the cosine of the angle $\theta$ between the field lines and the normal to the surface: $\Phi = \mathbf{B} \cdot A \cdot \cos \theta$
Mathematical Description
The magnetic field $\mathbf{B}$ can be described mathematically using several laws and equations:
Biot-Savart Law:
This law provides the magnetic field $\mathbf{B}$ produced by a small segment of current-carrying wire. For a current element $I \, d\mathbf{l}$ , the magnetic field at a point is given by: $d\mathbf{B} = \frac{\mu_0 I \, d\mathbf{l} \times \mathbf{r}}{4 \pi r^3}$ where $\mathbf{r}$ is the distance vector from the current element to the point, and $\mu_0$ is the permeability of free space ( $4 \pi \times 10^{-7}\ \text{T}\cdot\text{m/A}$ ).
Ampère’s Law:
This law relates the magnetic field around a closed loop to the current passing through the loop. It states that the line integral of the magnetic field $\mathbf{B}$ around a closed path is equal to $\mu_0$ times the current $I$ enclosed by the path: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$
Faraday’s Law of Induction:
This law describes how a changing magnetic field induces an electric field. The electromotive force (EMF) induced in a loop is proportional to the rate of change of the magnetic flux through the loop: $\mathcal{E} = -\frac{d\Phi}{dt}$
Magnetic Field Interactions
Forces on Moving Charges:
A charged particle moving in a magnetic field experiences a force known as the Lorentz force. This force is perpendicular to both the velocity of the particle and the magnetic field, and its magnitude is given by: $F = q v B \sin \theta$ where $q$ is the charge, $v$ is the velocity, $B$ is the magnetic field strength, and $\theta$ is the angle between the velocity and the magnetic field.
Forces on Currents:
A current-carrying conductor placed in a magnetic field experiences a force given by: $F = I L B \sin \theta$ where $I$ is the current, $L$ is the length of the conductor in the field, and $\theta$ is the angle between the conductor and the magnetic field.
Applications of Magnetic Fields
Electric Motors:
Magnetic fields are fundamental to the operation of electric motors. The interaction between the magnetic field generated by current-carrying coils and permanent magnets produces rotational motion.
Transformers:
Transformers use magnetic fields to transfer electrical energy between circuits through electromagnetic induction. A changing current in one coil generates a changing magnetic field that induces a voltage in another coil.
Magnetic Resonance Imaging (MRI):
MRI uses strong magnetic fields and radio waves to create detailed images of the inside of the body. The magnetic field aligns the nuclear spins in the body, and radio waves are used to perturb these spins and detect the resulting signals.
Electromagnetic Shielding:
Magnetic fields are used in electromagnetic shielding to protect sensitive electronic equipment from external magnetic interference. Shielding materials block or redirect unwanted magnetic fields.
Data Storage:
Magnetic fields are used in data storage devices, such as hard drives and magnetic tapes. Data is encoded as magnetic patterns on a storage medium, which can be read and written using magnetic heads.

Example 1

A proton (charge

q = 1.6 \times 10^{-19}\ \text{C}

) is moving with a velocity of

2 \times 10^6\ \text{m/s}

perpendicular to a magnetic field of strength

0.5\ \text{T}

. Calculate the magnetic force acting on the proton.

Example 2

A rectangular loop of dimensions

0.3\ \text{m} \times 0.4\ \text{m}

is placed in a uniform magnetic field of

0.2\ \text{T}

. If the magnetic field is perpendicular to the plane of the loop, calculate the magnetic flux through the loop.

Example 3

A straight wire of length

0.5\ \text{m}

carries a current of

3\ \text{A}

and is placed in a magnetic field of

0.4\ \text{T}

. Calculate the force on the wire if the magnetic field is perpendicular to the wire.

The force $F$ on a current-carrying wire in a magnetic field is given by:

$F = I L B \sin \theta$

where:

$I$ is the current ( $3\ \text{A}$ ),
$L$ is the length of the wire ( $0.5\ \text{m}$ ),
$B$ is the magnetic field strength ( $0.4\ \text{T}$ ),
$\theta$ is the angle between the wire and the magnetic field (here, $\theta = 90^\circ$ ).

Since $\sin 90^\circ = 1$ :

$F = I L B$

Substitute the given values:

$F = 3 \times 0.5 \times 0.4$

$F = 0.6\ \text{N}$

So, the force on the wire is $0.6\ \text{N}$ .

Problem 4: Magnetic Field at the Center of a Circular Loop

Problem: A circular loop of radius $0.1\ \text{m}$ carries a current of $2\ \text{A}$ . Calculate the magnetic field strength at the center of the loop.

Solution:

The magnetic field $B$ at the center of a circular loop is given by:

$B = \frac{\mu_0 I}{2 R}$

where:

$\mu_0$ is the permeability of free space ( $4 \pi \times 10^{-7}\ \text{T}\cdot\text{m/A}$ ),
$I$ is the current ( $2\ \text{A}$ ),
$R$ is the radius of the loop ( $0.1\ \text{m}$ ).

Substitute the given values:

$B = \frac{4 \pi \times 10^{-7} \times 2}{2 \times 0.1}$

$B = \frac{8 \pi \times 10^{-7}}{0.2}$

$B = \frac{4 \pi \times 10^{-6}}{0.1}$

$B = 4 \pi \times 10^{-5}$

Using $\pi \approx 3.14$ :

$B \approx 4 \times 3.14 \times 10^{-5}$

$B \approx 1.256 \times 10^{-4}\ \text{T}$

So, the magnetic field strength at the center of the loop is approximately $1.256 \times 10^{-4}\ \text{T}$ .

Problem 5: Induced EMF in a Moving Conductor

Problem: A straight conductor of length $0.2\ \text{m}$ moves with a velocity of $5\ \text{m/s}$ perpendicular to a magnetic field of $0.3\ \text{T}$ . Calculate the induced EMF in the conductor.

Solution:

The induced EMF $\mathcal{E}$ in a moving conductor is given by:

$\mathcal{E} = B v L$

where:

$B$ is the magnetic field strength ( $0.3\ \text{T}$ ),
$v$ is the velocity of the conductor ( $5\ \text{m/s}$ ),
$L$ is the length of the conductor ( $0.2\ \text{m}$ ).

Substitute the given values:

$\mathcal{E} = 0.3 \times 5 \times 0.2$

$\mathcal{E} = 0.3 \times 1$

$\mathcal{E} = 0.3\ \text{V}$

So, the induced EMF in the conductor is $0.3\ \text{V}$ .

Problem 6: Magnetic Force on a Charged Particle in a Uniform Magnetic Field

Problem: An electron (charge $q = 1.6 \times 10^{-19}\ \text{C}$ ) moves in a uniform magnetic field of $0.1\ \text{T}$ with a velocity of $1 \times 10^7\ \text{m/s}$ making an angle of $30^\circ$ with the magnetic field. Calculate the magnitude of the magnetic force on the electron.

Solution:

The magnetic force $F$ on a charged particle moving at an angle $\theta$ with the magnetic field is given by:

$F = q v B \sin \theta$

where:

$q$ is the charge of the electron ( $1.6 \times 10^{-19}\ \text{C}$ ),
$v$ is the velocity ( $1 \times 10^7\ \text{m/s}$ ),
$B$ is the magnetic field strength ( $0.1\ \text{T}$ ),
$\theta$ is the angle between the velocity and the magnetic field ( $30^\circ$ ).

Since $\sin 30^\circ = 0.5$ :

$F = q v B \sin 30^\circ$

Substitute the given value

$F$

$F = 0.6\ \text{N}$

So, the force on the wire is $0.6\ \text{N}$ .

Exercise &&1&& (&&1&& Question)

Exercise &&2&& (&&1&& Question)

U3AOS2 Topic 6: Magnetic Fields

Problem 4: Magnetic Field at the Center of a Circular Loop

Problem 5: Induced EMF in a Moving Conductor

Problem 6: Magnetic Force on a Charged Particle in a Uniform Magnetic Field

A circular loop of radius 0.1 m0.1\ \text{m}0.1 m carries a current of 2 A2\ \text{A}2 A. Calculate the magnetic field strength at the center of the loop.

A straight conductor of length 0.2 m0.2\ \text{m}0.2 m moves with a velocity of 5 m/s5\ \text{m/s}5 m/s perpendicular to a magnetic field of 0.3 T0.3\ \text{T}0.3 T. Calculate the induced EMF in the conductor.

A circular loop of radius $0.1\ \text{m}$ carries a current of $2\ \text{A}$ . Calculate the magnetic field strength at the center of the loop.

A straight conductor of length $0.2\ \text{m}$ moves with a velocity of $5\ \text{m/s}$ perpendicular to a magnetic field of $0.3\ \text{T}$ . Calculate the induced EMF in the conductor.