Magnetic Field Produced by Electric Currents Revision Notes

In these revision notes for Magnetic Field Produced by Electric Currents, we cover the following key points:

• How do electricity and magnetism interact with each other?
• What is the quantity used to represent the magnetic field? What kind of quantity is it (vector or scalar)?
• What are the factors affecting the magnetic induction? How to measure it?
• How to use the right-hand rule to determine the direction of magnetic field?
• How to represent symbolically the direction of magnetic field or that of current In 2D?
• How to find the direction and magnitude of a magnetic field produced by a long current-carrying wire?
• How to find the direction and magnitude of a magnetic field produced by a loop/ by a coil composed by a number of loops?
• How to find the direction and magnitude of a magnetic field produced by a solenoid?
• What is permeability of free space (vacuum)?
• What is relative permeability? How does it affect the properties of magnetic materials?
• How do we classify the magnetic materials?

Magnetic Field Produced by Electric Currents Revision Notes

An electric current produces a magnetic field in the space around it and magnetism generates electricity when a magnet is moving towards to or away from a coil in which a current Is flowing. These two properties form the foundations of electromagnetism.

Magnetic induction is the quantity that represents the magnetic field. It is analogue to the electric field E or gravitational field g. Magnetic induction is a vector quantity and is denoted by B⃗ in formulae. It depends on three factors:

• The amount of current I flowing through the conducting wire when magnetic field is generated,
• The force F by which we move the magnet towards or away from the coil (greater the force, stronger the magnetic field produced), and
• The length L of the conducting wire (magnetic induction is inversely proportional to the length of conductor for a constant force and current)

Mathematically, the magnetic induction is calculated by the formula

The unit of magnetic induction B known as Tesla, T. 1[T] = 1[N/A ∙ m]. In SI units, 1[T] = 1[kg/(A ∙ s² )].

The direction of magnetic field is found by using the right hand rule. It consists on grasping the wire by the right hand where the thumb lies in the direction of current flow. The other four fingers show the direction of magnetic field.

The formula of magnetic induction (in scalar form) produced by a long current carrying wire is

where μ₀ = 4π × 10^-7 N/A² is known as the magnetic permeability of free space (vacuum) and r is the distance from the wire.

Since it is impossible to plot a 3-D figure every time we have to deal with magnetic fields, we use a simpler notation to represent the direction of current when the situation is viewed from up. Thus, when the current enters the paper we use a x-symbol inside a circle while when the current Is flowing out of paper, we use a dot inside a circle to represent the direction of current. The magnetic field is easier when using such notation, as we have only to write concentric circles to represent it.

Since the intensity of magnetic field decreased with the increase in distance from the current carrying wire, we also decrease the intensity of magnetic field lines when representing it visually to give the idea of a weaker magnetic field when moving away from the wire.

The net magnetic field produced by two or more magnetic fields is the vector sum of all individual fields.

If a current flows in a circular loop, the four curled fingers are placed in the direction of electric current while the thumb shows the direction of magnetic field. Its magnitude is given by the formula:

where r is the radius of the loop. If the loop has many turns (N turns), we obtain a flat coil with a single radius. Thus, we obtain for the magnetic field at centre of coil:

The magnitude of magnetic induction (field) inside a solenoid (a spring-like conducting wire) of length L containing N turns when a current I flows through it, is

If we write n instead of N/L, where n represents the number of turns per unit length, the above equation becomes

The above equation is a demonstration that magnetic field inside a solenoid is directly proportional to the number of turns per unit length and the amount of current flowing through the solenoid.

The presence of extra matter, which affects the magnitude of magnetic field is represented through a quantity known as relative permeability, μr, which is calculated by

where B is the magnetic field in presence of a certain matter, B₀ is the corresponding magnetic field in vacuum; μ and μ0 are the values of permeability in the presence of the given matter and in vacuum respectively.

There are three types of magnetic materials:

1. Diamagnetic materials. They have a relative permeability slightly smaller than 1. When a diamagnetic is placed inside a magnetic field, it is weakly magnetised in the opposite direction of magnetic field. As a result, the overall magnetic field decreases and the magnetic field lines diverge from each other when approaching the material. Carbon, bismuth, silver and copper are examples of diamagnetic materials.
2. Paramagnetic materials. This category includes materials that have a relative permeability slightly higher than 1. Examples of paramagnetic materials include aluminium, magnesium, air, etc.
   When a diamagnetic is placed inside a magnetic field, it is slightly magnetised in the direction of this field. Therefore, the magnetic field lines slightly converge when approaching the material.
3. Ferromagnetic materials are those materials that have their relative permeability much greater than 1 (a few hundred to a few thousand times greater than 1). They are strongly magnetised when placed inside a magnetic field. As a result, the magnetic field enforces and the field lines converge when approaching the material.
   

   Iron cobalt and nickel are examples of ferromagnetic materials. Due to their strong magnetic properties, ferromagnetic materials are used to produce artificial magnets.