Electromagnetism

Electromagnetism is among the forces of nature that deal with the study of how electric charges and magnetic fields interact with each other. Being an important branch of physics, it finds numerous technological applications as well as helps to explain various natural phenomena.

1.0Electromagnetism Explained

Key Concepts

To better understand electromagnetism, we need to explore some basic concepts:

1. Electric Charge(q): 

It is one of the fundamental properties of matter, and there exist two kinds of electric charges: positive and negative. Opposite charges attract, whereas like charges repel. This force between charges is called the electrostatic force, and it forms one of the basic principles in classical electromagnetism.

2. Electric Field: 

An electric field is the space surrounding a charged particle in which its electric force can be felt by other charges. The direction of the electric field is from positive to negative charge. The strength of the given electric field is directly proportional to the voltage (V) & inversely proportional to the distance present between the charges. Its unit is represented by Volt per metre or Coulomb per sq metre. The general formula for the electric field is:

$E=\frac{F}{q}$

Here:

• E = Electric field (in volts per meter, V/m)
• F = Force experienced by the test charge (in newtons, N)

A test charge is a small charge used to measure the electric field at a given point in space without disturbing the field itself. It is assumed to be so small that it does not influence or alter the electric field that is being measured.

• q = Magnitude of the test charge (in coulombs, C)

3. Magnetic Force: 

The Magnetic Force is a force experienced by a moving charge in a magnetic field. It can be calculated using different formulas depending on the situation. Some of them are: 

Magnetic Force on a Moving Charge:

$F=q(v\times B)$

Here, 

F = magnetic force

q = electric charge. 

v = velocity of the charge. 

B = Magnetic field. 

$\theta$ = Angle between the velocity and magnetic field

Magnetic Force on a Current-Carrying Wire:

$F=ILBsin\theta$

Here:

• I = Current in the wire (in amperes, A)
• L = Length of the wire segment within the magnetic field (in meters, m)

Force Between Two Parallel Current-Carrying Wires:

Two parallel wires carrying current experience a magnetic force due to the interaction of their magnetic fields. The force per unit length between two parallel wires separated by a distance r is given by:

$F_{Perunitlength}=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$

Here:

• μ₀ = Permeability of free space (4π×10⁻⁷ N/A)
• I₁​ and I₂​ = Currents in the two wires (in amperes, A)

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

If a charged particle moves through a uniform magnetic field at an angle, it follows a circular or helical path under the influence of the magnetic force. The radius r of the circular motion is determined by:

$r=\frac{mv}{qBsin\theta }$

Here:

• m = Mass of the charged particle (in kilograms, kg)
• v = Speed of the particle (in meters per second, m/s)

4. Magnetic Field(B): 

A magnetic field is a space around a magnetic material or moving electric charge where the force of magnetism can be observed. Magnetism is one of the fundamental phenomena of electromagnetism, which is that a magnetic field is induced around any moving charge. The direction of the magnetic field at any point in space is determined by the right-hand rule for various forms of current or charges.

Right-Hand Rule for a Wire: Extend your right thumb in the direction of the current. Then, curl your fingers around the wire. The direction that your fingers curl is the direction of the magnetic field lines around the wire.

Electromagnetism and Magnetism

Electromagnetism and magnetism are two phenomena that are related to each other. Magnetism is the property of a material which reacts to the magnetic field. Whenever electricity passes through a wire, then a magnetic field will develop around it. A magnetic field can also be induced to pass through a conductor by an electric current (electromagnetic induction). The study of electricity and magnetism together constitutes the foundation of electromagnetism.

Electromagnetism and Electromagnetic Induction

Electromagnetism and electromagnetic induction are closely related. Electromagnetic induction is the specific phenomenon of inducing an electric current by a changing magnetic field (B). This phenomenon was discovered by the renowned physicist Michael Faraday and serves as the foundation for many electrical devices today, such as transformers and generators.

2.0Faraday’s Law of Electromagnetism

Faraday's law of induction is one of the main principles of electromagnetism, and it explains how one can induce an electric current in the presence of a magnetic field. The British scientist Michael Faraday discovered the law in the 19th century and provided the founding stone for modern electrical engineering.

Statement of Faraday's Law:

Faraday's Law states that the EMF induced in a coil or circuit is proportional to the change in magnetic flux passing through it.

Mathematically, Faraday's Law is stated to be:

$\epsilon =-\frac{d\varphi }{dt}$

Here, 

$\epsilon$ is the induced EMF (electromotive force) 

$\varphi$ is the magnetic flux.

$\frac{d\varphi }{dt}$ is the rate of magnetic flux. 

The negative sign indicates that the induced EMF opposes the change in magnetic flux. This is the application of Lenz's Law, which states that the direction of the induced current results from a changing magnetic field.

3.0Applications of Faraday’s Law

1. Electrical Generators: The operation of a generator applies the idea of changing a magnetic field in the space that passes over a coil where there will be rotation through the coil, inducing electricity through changing magnetic flux in that space. 
2. Transformers: The workability of transformers depends upon Faraday's Law of Induction. When a change in current occurs within the primary coil, the secondary coil experiences an altering magnetic flux, thereby producing EMF.
3. Induction Cooktops: In induction cooking, alternating magnetic fields induce electric currents in the cookware, which then generates heat due to electrical resistance.

4.0Solved Examples

Problem 1: A circular loop of wire with a radius of 0.1 m carries a current of 2 A. Determine the magnetic field (B) at the centre of the loop.

Solution: The magnetic field at the centre of a circular current loop is given by:

$B=\frac{{\mu }_{0}I}{2.R}$

Where:

B= Magnetic field at the centre of the loop (in Teslas)

${\mu }_0=4\pi \times 10^{-7}Tm/A$=permeability of free space,

I=2A=current,

R=0.1m=radius of the loop.

Substitute the given values:

$B=\frac{4\pi \times 10^{-7}\times 2}{2\times 0.1}=\frac{8\pi \times 10^{-7}}{0.2}=1.257\times 10^{-6}T$

Thus, the magnetic field at the centre of the loop is approximately $1.26\mu T$.

Problem 2: A solenoid has 1500 turns of wire and a length of 0.8 meters. With a current of 3 A flowing through the solenoid, calculate the magnetic field inside the solenoid.

Solution: The magnetic field inside a solenoid is given by:

$B={\mu }_0.n.l$

Where:

B = magnetic field(in teslas),

${\mu }_0=4\pi \times 16^{-7}$Tm/A,

n=$\frac{N}{L}$= number of turns per unit length,

N= 1500 turns(total number of turns),

L= 0.8 m (length of the solenoid),

I= 3A (current).

First, calculate the number of turns per unit length:

$n=\frac{1500}{0.8}$=1875 turns/m

Now substitute the values into the formula for the magnetic field:

$B=4\pi \times 10^{-7}\times 1875\times 3$

$B=7.065\times 10^{-3}T$

Thus, the magnetic field inside the solenoid is approximately 7.07 mT.

Problem 3: A proton moves through a magnetic field in a circular path. The radius of the circular path is 0.2 m, and the speed of the proton is unknown. The magnetic field has a strength of 0.4 T. The proton experiences a magnetic force of 3.2×10⁻¹³ N. Find the speed of the proton.

Solution: 

Given Radius r = 0.2m, B = 0.4T

Charge on a proton = $1.6\times 10^{-19}$

Mass of a proton = $1.67\times 10^{-27}$

As a charged particle moves in a circular path in a magnetic field, the magnetic force supplies the centripetal force, which keeps it in circular motion. This provides the following relation:

$F=\frac{m{v}^2}{r}$

And at the same time, the magnetic force on any moving charged particle is given by: 

$F=qvB$

So, 

$\frac{m{v}^2}{r}=qvB$

$v=\frac{qBr}{m}$

$v=\frac{(1.6\times 10^{-19})(0.4)(0.2)}{1.67\times 10^{-27}}$

$v=\frac{1.28\times 10^{-19}}{1.67\times 10^{-27}}$

$v=7.65\times 10^{7}m/s$