Magnetic Field Lines
Patterns Around Magnets
Magnetic field lines visually represent the direction and strength of a magnetic field.
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.
- 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.
Magnetic Field around a Bar Magnet
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.
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).
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.
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: $$F = qvB \sin \theta$$ 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).
- $\theta$ is the angle between the velocity vector and the magnetic field vector.
If a proton ($q = 1.6 \times 10^{-19} \, \text{C}$) moves at $2 \times 10^6 \, \text{m s}^{-1}$ through a magnetic field of $0.5 \, \text{T}$ at an angle of $90^\circ$, the force is:
$$F = (1.6 \times 10^{-19} \, \text{C})(2 \times 10^6 \, \text{m s}^{-1})(0.5 \, \text{T}) \sin 90^\circ $$
$$= 1.6 \times 10^{-13} \, \text{N}$$
Students often forget that the magnetic force is zero if the velocity is parallel to the magnetic field ($\theta = 0^\circ$ or $180^\circ$).
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: $$F = BIL \sin \theta$$ 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.
- $\theta$ is the angle between the current direction and the magnetic field.
Consider a wire carrying a current of $3 \, \text{A}$ in a magnetic field of $0.4 \, \text{T}$. If the wire is $0.5 \, \text{m}$ long and the angle between the current and the field is $60^\circ$, the force is:
$$F = (0.4 \, \text{T})(3 \, \text{A})(0.5 \, \text{m}) \sin 60^\circ $$
$$= 0.52 \, \text{N}$$
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: $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi r}$$ where:
- $F$ is the force between the wires.
- $L$ is the length of the wires.
- $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \, \text{T m A}^{-1}$).
- $I_1$ and $I_2$ are the currents in the wires.
- $r$ is the distance between the wires.
If two wires, each carrying a current of $5 \, \text{A}$, are separated by $0.1 \, \text{m}$, the force per unit length is:
$$\frac{F}{L} = \frac{(4\pi \times 10^{-7} \, \text{T m A}^{-1})(5 \, \text{A})(5 \, \text{A})}{2 \pi (0.1 \, \text{m})}$$
$$ = 5 \times 10^{-6} \, \text{N m}^{-1}$$
The force between the wires is attractive if the currents flow in the same direction and repulsive if they flow in opposite directions.
- 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 $\theta$ between its velocity and the magnetic field is $0^\circ$?
- How does the direction of current affect the force between two parallel wires?
