Magnetic Flux Density Revision Notes for Leaving Cert Physics

Magnetic Flux Density

Understanding magnetic flux density

Magnetic flux density is a fundamental concept that helps us understand how magnetic fields interact with electric currents. When physicists want to describe the strength of a magnetic field and its effect on current-carrying conductors, they use a quantity called magnetic flux density, represented by the symbol B.

The magnetic flux density at any point in a magnetic field tells us exactly how strong the magnetic effect is at that location. Think of it as a way to measure the "intensity" of the magnetic field. More specifically, magnetic flux density is defined as the force that would act on a standard test conductor placed in the field.

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Formal Definition of Magnetic Flux Density

Magnetic flux density (B) at a point in a magnetic field is a vector quantity whose magnitude equals the force experienced by a conductor of length 1 metre carrying a current of 1 ampere when placed at right angles to the field at that point.

Since magnetic flux density is a vector quantity, it has both magnitude and direction. The direction of the magnetic flux density is determined by the direction of force that would act on a north magnetic pole if placed at that point in the field.

The tesla unit

The SI unit for measuring magnetic flux density is the tesla, symbolised as T. This unit is named after Nikola Tesla (1856-1943), a Serbian-American engineer and inventor who made groundbreaking contributions to the design of AC power supply systems and invented the Tesla coil.

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About the Tesla Unit

The tesla is defined in a very practical way: the magnetic flux density at a point is 1 tesla if a conductor of length 1 metre carrying a current of 1 ampere experiences a force of 1 newton when placed perpendicular to the magnetic field.

This definition gives us a clear, measurable way to determine magnetic flux density using quantities we can easily measure: force, current, and length.

Mathematical relationship and formula

The relationship between magnetic flux density, current, length, and force is expressed by the fundamental equation:

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The Fundamental Formula

$F = BIL$

Where:

F = force on the conductor (in newtons)
B = magnetic flux density (in tesla)
I = current through the conductor (in amperes)
L = length of conductor in the magnetic field (in metres)

This formula can be rearranged depending on what quantity you need to find:

To find magnetic flux density: $B = \frac{F}{IL}$
To find current: $I = \frac{F}{BL}$
To find length: $L = \frac{F}{BI}$

Worked calculations and examples

Let's look at some practical calculations to understand how to use these concepts:

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Example 1: Finding Magnetic Flux Density

A straight piece of wire 3 metres long carries a current of 2 amperes. When placed perpendicular to a uniform magnetic field, it experiences a force of 12 newtons. What is the magnetic flux density of the field?

Solution: Using the formula: $B = \frac{F}{IL}$

Substituting the values: $B = \frac{12}{2 \times 3} = \frac{12}{6} = 2 \text{ T}$

Answer: The magnetic flux density is 2 tesla.

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Example 2: Calculating Force on a Conductor

A conductor of length 40 centimetres carries a current of 3 amperes and is placed perpendicular to a uniform magnetic field with flux density 5.2 tesla. Calculate the force acting on the conductor.

Solution: First, convert length to metres: 40 cm = 0.4 m

Using: $F = BIL$

Substituting: $F = 5.2 \times 3 \times 0.4 = 6.24 \text{ N}$

Answer: The force acting on the conductor is 6.24 N.

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Example 3: Current Loop in Magnetic Field

Consider a rectangular loop of wire with dimensions 20 cm × 12 cm carrying a current of 4 amperes in a uniform magnetic field of 0.6 tesla.

Solution: For the 20 cm sides of the loop: $F = BIL = 0.6 \times 4 \times 0.2 = 0.48 \text{ N}$

Result: The forces on opposite sides of the loop are equal and opposite, but since the loop is free to rotate about its axis, these forces create a turning effect that causes the coil to rotate.

Base SI units

Understanding how the tesla relates to fundamental SI units is important for dimensional analysis and unit conversions.

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Derivation of Tesla in Base SI Units

We can derive this by starting with the force equation:

From $F = BIL$ , we get $B = \frac{F}{IL}$

Substituting the base SI units:

Force (F) is measured in newtons: 1 N = 1 kg⋅m⋅s⁻²
Current (I) is measured in amperes: A
Length (L) is measured in metres: m

Therefore: $B = \frac{\text{kg⋅m⋅s⁻²}}{\text{A⋅m}} = \text{kg⋅s⁻²⋅A⁻¹}$

Result: 1 tesla = 1 kg⋅s⁻²⋅A⁻¹

Applications in current loops

When current flows through a loop or coil in a magnetic field, the magnetic flux density determines the forces acting on different parts of the loop. These forces can cause the loop to rotate, which is the basic principle behind electric motors.

In a rectangular current loop, the forces on opposite sides are equal in magnitude but opposite in direction. When the loop is free to rotate, these forces create a couple that causes rotational motion. This rotation continues as long as current flows through the loop in the magnetic field.

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Motor Principle

The rotation of current loops in magnetic fields forms the fundamental operating principle of electric motors. The continuous rotation is achieved by switching the current direction at appropriate times, maintaining the rotational motion.

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Key Points to Remember:

Magnetic flux density (B) measures the strength of a magnetic field and is measured in tesla (T)
The key formula is F = BIL for force on a current-carrying conductor perpendicular to a magnetic field
1 tesla is the magnetic flux density when 1 metre of conductor carrying 1 ampere experiences 1 newton of force
Tesla in base SI units: 1 T = 1 kg⋅s⁻²⋅A⁻¹
Current loops in magnetic fields experience forces that can cause rotation, forming the basis of electric motors

Magnetic Flux Density (Leaving Cert Physics): Revision Notes