Magnetic Flux Density: MDCAT Physics notes
Magnetic Flux Density MDCAT notes: definition of B, the tesla and gauss, field lines, field of a wire and a solenoid, B = μ₀nI, and parallel currents.
Source of magnetic field
A magnetic field is produced by moving charges, that is by electric currents. A current loop is the simplest source; a bar magnet's field comes from the motion of electrons in its atoms. A static charge produces no magnetic field, and isolated magnetic poles (monopoles) do not exist. Cutting a magnet in two gives two smaller magnets, each a dipole with its own N and S poles.
Magnetic flux density
The force on a straight conductor of length $L$ carrying current $I$ at angle $\theta$ to a uniform field is $F = BIL\sin\theta$. The magnetic flux density (magnetic field strength, magnetic induction) is defined from this:
$$B = \frac{F}{IL\sin\theta} \qquad \text{or } B = \frac{F}{IL}\ \text{when } \theta = 90^\circ$$
- $B$ is a vector.
- SI unit: tesla. 1 T = 1 N A$^{-1}$ m$^{-1}$ = 1 Wb m$^{-2}$.
- $B$ is also the flux per unit area held at right angles to the field, so it is called flux density.
- Smaller unit: 1 gauss (G) = $10^{-4}$ T.
Magnetic field lines
- Outside a magnet they run from the N pole to the S pole; inside they run S to N, forming closed loops.
- They never cross or touch each other.
- The field is strongest where lines are closest together.
- Between like poles the lines push apart (repel), leaving a neutral point.
Fields of currents
| Conductor | Shape of field lines | Strength |
|---|---|---|
| Long straight wire | Concentric circles around the wire | $B = \dfrac{\mu_0I}{2\pi r}$ |
| Long solenoid (inside) | Straight, parallel to the axis | $B = \mu_0nI$, uniform |
$\mu_0 = 4\pi\times10^{-7}$ Wb A$^{-1}$ m$^{-1}$ and $n$ = number of turns per unit length. The direction is given by the right-hand grip rule. In each case $B \propto I$: doubling the current doubles the field.
Cutting a solenoid
If a solenoid is cut into pieces (in any ratio) and each is connected to the same current, every piece has the same turns per metre $n$, so $B = \mu_0nI$ is the same in each.
Force on a conductor in a field
$F = BIL\sin\theta$ is zero when the conductor is parallel to the field and maximum ($BIL$) when perpendicular. Example: a 2 m wire with 3 A lying along a field of 5 G experiences zero force.
Parallel currents
Each wire lies in the magnetic field of the other, so a magnetic force acts:
- Currents in the same direction attract.
- Currents in opposite directions repel.
The wires are electrically neutral, so the force is magnetic, not electric; the electric field between them is unaffected.
Key formulas
- $F = BIL\sin\theta$
- $B = \mu_0I/2\pi r$ (straight wire)
- $B = \mu_0nI$ (solenoid)
- 1 T = 1 Wb m$^{-2}$ = 1 N A$^{-1}$ m$^{-1}$ = $10^4$ G
Common MDCAT traps
- The unit of $B$ is Wb m$^{-2}$, not Wb m$^{2}$ or Wb.
- Field is strongest where lines are crowded, not spread out.
- Cutting a solenoid does not change $B$ in each part; $n$ stays the same.
- A static charge or an isolated pole is not a source of magnetic field.
- Parallel like currents attract (opposite to like charges).
Quick revision
- Magnetic field strength = magnetic induction = flux density, unit tesla.
- Field inside a long solenoid is uniform and axial.
- Field lines around a straight wire are circles.
- Broken magnet pieces each have N and S poles.