Electromagnetism Study Notes

CHAPTER FIVE: ELECTROMAGNETISM

5.1 Magnetic Field due to an Electric Current

Overview

Magnetic fields can be generated not only by permanent magnets but also by electric currents. An experiment involves a piece of wire passing vertically through a horizontal sheet of cardboard sprinkled with iron filings. When a current is passed through the wire, the filings align to form a circular field pattern with the wire at the center. The pattern can be seen after gently tapping the cardboard, and the direction of the flux lines can be determined using a compass.

Observations

1. Change with Current Direction: If the current direction is reversed, so too is the direction of the lines of flux.

2. Effect of Current: The effect on the filings and compass needle disappears when the current is turned off. Hence, the magnetic field is produced by the electric current.

3. Field Strength Dependence: The strength of the magnetic field is directly proportional to the current; thus, increasing the current increases field strength. Furthermore, the field strength diminishes with distance from the current-carrying conductor.

Magnetic Field Patterns

The investigation of the entire length of the conductor reveals that the magnetic field surrounding a straight conductor is in concentric cylindrical forms.

Standard Conventions

In depicting electric current flow:

• Current flowing into the paper (away from the viewer) is represented by a specified symbol (often an arrow's feathered end).

• Current flowing out of the paper (toward the viewer) is indicated by a dot (the arrow's point).

Direction of Magnetic Lines of Flux

The screw rule serves as a mnemonic for remembering the direction of the magnetic lines of flux: “If a right-hand thread screw is turned in the direction of the current along the conductor, the direction of the rotation of the screw indicates the magnetic field’s direction.” For example, current flowing away from the viewer would require a clockwise rotation direction of the screw, establishing that the magnetic field is clockwise.

5.2 Electromagnets

Importance of Solenoids

Solenoids are crucial in electromagnetic theory due to the uniform magnetic field they create inside, which can be modified by varying the current through the solenoid.

Applications of Electromagnets

1. Electric Bells: Utilize the attraction exerted by an electromagnet on a soft iron armature.

   • Single Stroke Bell: In a circuit, pressing a push button activates the current, energizing the coil and attracting the armature (which moves to strike a gong). If current ceases, the coil demagnetizes, and the armature returns to its resting position.

   • Circuit Representation: Illustrated with the corresponding circuitry.

2. Relays: Similar to electric bells, but instead of striking a gong, they open/close electrical contacts.

   • The energized coil attracts a hinged iron armature, connecting fixed contacts and closing another circuit.

3. Lifting Magnets: Employed in metal works, these incorporate large electromagnets designed to lift scrap metal.

   • Illustrated with an elevation and plan view to indicate the coil (C) wound around the iron core (F).

   • When energized, the coil lifts magnetic materials.

4. Telephone Receivers: Function by converting electrical waves back into sound waves.

   • Typically consists of a permanent magnet with coils on its poles, causing a diaphragm to vibrate and produce sound.

5.3 Force on a Current-Carrying Conductor

Interaction of Magnetic Fields

When a current-carrying conductor is placed in the magnetic field of permanent magnets, there is an interaction between the fields, resulting in a force acting on the conductor. Several factors influence this force:

1. The magnetic flux density, B (in teslas)

2. The magnitude of the current, I (in amperes)

3. The length of the conductor within the field, l (in meters)

4. The relative directions of the magnetic field and current.

Force Calculation Formulas

1. When current and magnetic field are perpendicular:
   $F = BIl$
   (where F is the force in newtons)

2. When at an angle θ:
   $F = BIl imes ext{sin}( heta)$

Definition of Magnetic Flux Density

The magnetic flux density (B) can also be defined from the above formula as:
$B = rac{F}{Il}$

Practical Example

When a specific current-carrying conductor is placed in a magnetic field, the two fields interact, exerting a force which tends to move the conductor downwards (taking an electric motor as an example).

Determining Force Direction

To determine force direction, use Fleming’s Left-Hand Rule (often termed the motor rule):

• The extended thumb, first finger, and second finger of the left hand must all be at right angles:

  • First finger: Direction of the magnetic field (B)

  • Second finger: Direction of the current (I)

  • Thumb: Direction of motion (F)

5.4 Force on a Charge

Charge in Magnetic Field Interaction

When a charge (Q coulombs) moves at velocity (v m/s) in a magnetic field with flux density (B teslas), the formula for force exerted on the charge is:
$F = QvB$

Example Problem

An example given involves an electron with charge $1.6 imes 10^{-19}$ coulombs traveling at $3 imes 10^7$ m/s in a magnetic field with flux density of $18.5 imes 10^{-6}$ T, leading to:
$F = (1.6 imes 10^{-19}) (3 imes 10^7) (18.5 imes 10^{-6}) = 8.88 imes 10^{-17} ext{ N}$

5.5 Problems

5.5.1 Force on a Current-Carrying Conductor

1. A 200 mm conductor carrying 70 A in a magnetic field of 1.5 T has a force of 21.0 N when at right angles. It exerts a force of 14.8 N when at 45°.

2. Current required in a 240 mm length of conductor for a force of 1.20 N in 1.25 T is calculated as 4.0 A.

3. For a 30 cm conductor in a 15 A field producing 3.6 N of force, the strength of the field is found to be 0.80 T.

4. A 300 mm conductor with a 13 A current in between circular poles of 80 mm diameter has a force of 0.582 N.

5. A 400 mm vertical conductor with current of 25 A has a force of 80 N, leading to pole face diameter determination.

   • (b) Direction of force is towards the viewer when the magnetic field is from left to right.

6. For a coil on a former, the force on each side is calculated for both a single-turn (2.25×10^-3 N) and a 400-turn coil (0.9 N).

5.5.2 Force on a Charge

7. Force on a charge $2 imes 10^{-18} C$ traveling in a field of $2 imes 10^{-7} T$ is calculated to be $8 imes 10^{-19} N$.

8. The speed of a charge of $10^{-19} C$ traveling in a field when the force is $10^{-20} N$ is $10^6 m/s$.