D.2.2 Magnetic field properties and laws

Magnetic Field Lines

Patterns Around Magnets

Magnetic field lines visually represent the direction and strength of a magnetic field.

Example

Magnetic field lines always point from the north pole to the south pole outside a magnet, and from south to north inside the magnet, forming closed loops.

Example

Consider a bar magnet:

If you sprinkle iron filings around it, the filings align along the magnetic field lines, revealing a pattern that is densest near the poles (where the field is strongest) and spreads out as you move away.

Patterns Around Wires

A straight current-carrying wire generates a magnetic field with lines forming concentric circles around the wire.

The right-hand rule helps determine the direction:

Point your thumb in the direction of the current.
Your fingers will curl in the direction of the magnetic field lines.

Example

If the current flows upward, the magnetic field lines will circle the wire in a counterclockwise direction.

Patterns Around Solenoids

A solenoid is a coil of wire that produces a magnetic field similar to a bar magnet when current flows through it.
Inside the solenoid, the field lines are parallel and uniform, indicating a strong and constant magnetic field.
Outside, the lines resemble those of a bar magnet, exiting from one end (the north pole) and entering the other (the south pole).

Tip

To find the direction of the magnetic field in a solenoid, use the right-hand grip rule:

Curl your fingers in the direction of the current, and your thumb will point toward the solenoid’s north pole.

Magnetic field lines around the solenoid. Using the right-hand grip rule, the thumb points toward left.

Force on a Moving Charge

When a charged particle moves through a magnetic field, it experiences a force called the magnetic force.
The magnitude of this force is given by the formula:

$F = q v B \sin θ$

where:

$F$ is the magnetic force.
$q$ is the charge of the particle.
$v$ is the velocity of the particle.
$B$ is the magnetic flux density (strength of the magnetic field).
$θ$ is the angle between the velocity vector and the magnetic field vector.

Example

Magnetic force on a moving charge

If a proton ( $q = 1.6 \times 10^{- 19} C$ ) moves at $2 \times 10^{6} m/s$ through a magnetic field of $0.5 T$ at an angle of $90^{\circ}$ , the force is:

$F = (1.6 \times 10^{- 19} C) (2 \times 10^{6} m/s) (0.5 T) \sin 90^{\circ}$

$= 1.6 \times 10^{- 13} N$

Common Mistake

Students often forget that the magnetic force is zero if the velocity is parallel to the magnetic field ( $θ = 0^{\circ}$ or $180^{\circ}$ ).

Tip

Use the right-hand rule to determine the direction of the force on a positive charge:

Point your fingers in the direction of the velocity ( $v$ ).
Align your palm with the magnetic field ( $B$ ).
Your thumb will point in the direction of the force ( $F$ ).For a negative charge, the force is in the opposite direction.

Force on a Current-Carrying Wire

A wire carrying an electric current in a magnetic field also experiences a force.
The magnitude of this force is given by:

$F = B I L \sin θ$

where:

$F$ is the magnetic force.
$B$ is the magnetic flux density.
$I$ is the current in the wire.
$L$ is the length of the wire in the magnetic field.
$θ$ is the angle between the current direction and the magnetic field.

Example

Magnetic force of a current-carrying wire

Consider a wire carrying a current of $3 A$ in a magnetic field of $0.4 T$ . If the wire is $0.5 m$ long and the angle between the current and the field is $60^{\circ}$ , the force is:

$F = (0.4 T) (3 A) (0.5 m) \sin 60^{\circ}$

$= 0.52 N$

Tip

To find the direction of the force, use the right-hand rule for currents:

Point your thumb in the direction of the current ( $I$ ).
Align your palm with the magnetic field ( $B$ ).
Your fingers will point in the direction of the force ( $F$ ).

Force Between Parallel Wires

Two parallel wires carrying currents exert forces on each other due to their magnetic fields.
The force per unit length between two wires is given by:

$\frac{F}{L} = \frac{μ_{0} I_{1} I_{2}}{2 π r}$

where:

$F$ is the force between the wires.
$L$ is the length of the wires.
$μ_{0}$ is the permeability of free space ( $4 π \times 10^{- 7} T m/A$ ).
$I_{1}$ and $I_{2}$ are the currents in the wires.
$r$ is the distance between the wires.

Illustration of the forces between two parallel wires.

Example

If two wires, each carrying a current of $5 A$ , are separated by $0.1 m$ , the force per unit length is:

$\frac{F}{L} = \frac{(4 π \times 10^{- 7} T m/A) (5 A) (5 A)}{2 π (0.1 m)}$

$= 5 \times 10^{- 6} N/m$

Hint

The force between the wires is attractive if the currents flow in the same direction and repulsive if they flow in opposite directions.

Self review

How do magnetic field lines differ around a bar magnet, a straight wire, and a solenoid?
What happens to the magnetic force on a charge if the angle $θ$ between its velocity and the magnetic field is $0^{\circ}$ ?
How does the direction of current affect the force between two parallel wires?

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D.2.2 Magnetic field properties and laws

Magnetic Field Lines

Patterns Around Magnets

Patterns Around Wires

Patterns Around Solenoids

Force on a Moving Charge

Magnetic force on a moving charge

Force on a Current-Carrying Wire

Magnetic force of a current-carrying wire

Force Between Parallel Wires

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Magnetic Fields