Electromagnetism in conduction, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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Foundation Electrical and

Electronic Engineering

N Drury [email protected]

Learning Objectives

• Recap flux, flux density and describe the shape of

different magnetic fields

• Describe the field around a toroidal coil

• Use Ampere’s Law to calculate the strength of a

magnetic field and understand the significance of

Magnetic Permeability

• Calculate the Lorentz Force for a current-carrying wire

in a B field

• Calculate the force on moving charges in a B field

• Magnetic Flux, j If Flux Density, B is a measure of the number of field lines cutting perpendicularly through a plane, then
• 𝐵 = 𝜑 𝐴 where j is the total number of lines of magnetic force in a material
• So j = BA
• Flux is measured in Weber (Wb). 1 Wb ≡ 1 Tm 2

Magnetic Quantities

Magnetic field around a straight wire

• A current-carrying wire generates a circular magnetic field around itself

Current conventions

• This symbol denotes the movement of charges, or a vector force, directly out of the page or screen
• This symbol denotes the movement of charges, or a vector force, directly into the page or screen

Magnetic field around a loop of wire

• Around a single coil the field lines from each part of the wire combine
• The right hand grip rule can be used to work out the resultant field direction

The Toroidal Coil

• A toroidal coil is simply a solenoid whose ends are bent together to form a continuous loop

The Toroidal Coil

• A toroid is more compact and lighter than other shaped cores
• Due to its symmetry, the amount of magnetic flux that escapes outside the core (flux leakage) is low; therefore it is more efficient and radiates less electromagnetic interference
• Because it is a closed-loop core it will have a higher magnetic field than a solenoid. This is because most of the magnetic field is contained within the core
• Toroidal inductors and transformers are used in: power supplies, inverters, and amplifiers, which in turn are used in electrical equipment such as TVs, radios, computers, and audio systems

Ampere’s Law

• The magnetic field around a conductor is proportional to the size of the current through it
• Ampere’s law states that for any closed loop path around a conductor, if we split the path into length elements Dl, the sum of the length elements of the path multiplied by the B field in the direction of the length element is equal to the permeability multiplied by the enclosed current

Ampere’s Law for a straight wire

• σ^ 𝐵∥Δ𝑙 = 𝜇𝑜𝐼
• For a straight wire, we know we get a circular field
• At distance r from wire, σ^ Δ𝑙 = 2 𝜋𝑟
• Giving us 2𝜋𝑟 B = 𝜇 𝑜

• Or 𝐵 = 𝜇𝑜𝐼 2 𝜋𝑟

Ampere’s Law for a solenoid

• If we consider a rectangular path with length L parallel to the coil, then the field is perpendicular to the sides of the path and if we make it so that the other parallel end is very far from the coil so the field is negligible, then Ampere’s Law gives us:
• BL = 𝜇 𝑜

• B =

𝜇𝑜𝑁𝐼 𝐿 = 𝜇𝑜𝑛𝐼 Where n is the number of turns / m

Problems

• What is the magnetic field at the centre of a 0.1 m long solenoid with a turn density = 100 turns/m if the applied current is equal to 20 A?
• What current would be required in the above solenoid to generate a 1T field?

Relative permeability

Solenoid with an iron core

• The magnetic permeability of iron is much higher than that of air
• An iron core has the effect of greatly multiplying the magnetic field of a solenoid