Biot–Savart Law & Field due to Currents

PHYMagnetic Field & Forces

The Biot–Savart law gives the magnetic field produced by a current element, and integrating it (or using Ampere's law as a shortcut) yields the standard results for a straight wire, a circular loop, a point on the loop's axis, a solenoid and a toroid. ISC numericals are almost always direct substitution into one of these standard formulas, so recognising the geometry is the whole skill.

Note that the field always wraps around the current (right-hand rule) and falls off with distance.

Biot–Savart law (magnitude)

dB = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,dl\,\sin\theta}{r^2}

dl

current-element length,

\theta

angle between the element and

\vec{r}

r

distance to the field point,

\mu_0/4\pi = 10^{-7}\ \mathrm{T\,m\,A^{-1}}

Long straight wire

B = \dfrac{\mu_0 I}{2\pi a}

a

perpendicular distance from the wire; valid for an infinitely long straight conductor.

Centre of a circular coil (

N

turns)

B = \dfrac{\mu_0 N I}{2R}

R

radius of coil,

N

number of turns, field directed along the axis through the centre.

Axial field of a circular coil

B = \dfrac{\mu_0 N I R^2}{2\,(R^2 + x^2)^{3/2}}

x

distance of the point from the centre along the axis; reduces to

\mu_0 NI/2R

when

x=0

Inside a long solenoid

B = \mu_0 n I

n = N/L

turns per unit length; field is uniform along the axis inside the solenoid.

Inside a toroid

B = \dfrac{\mu_0 N I}{2\pi r}

N

total turns,

r

mean radius of the toroid; field is confined to the core and zero outside.

Use $\mu_0/4\pi = 10^{-7}\ \mathrm{T\,m\,A^{-1}}$ in Biot–Savart; use $\mu_0 = 4\pi\times10^{-7}$ in the loop/solenoid formulas.
For a circular coil always include the number of turns $N$ ; a single loop is the $N=1$ case.
The solenoid result $B=\mu_0 nI$ uses $n=N/L$ (turns per metre), not the total turns $N$ .
The toroid uses the mean radius $r$ and total turns $N$ ; the field outside both the core and the toroid is zero.
All lengths must be converted to metres ( $\mathrm{cm}\to\times10^{-2}\ \mathrm{m}$ ) before substituting.
Fields from several wires add as vectors; if two fields are mutually perpendicular the resultant is $\sqrt{B_1^2+B_2^2}$ .
The straight-wire field circles the conductor; direction is given by the right-hand grip rule (thumb along $I$ , fingers along $\vec{B}$ ).

Where the marks go

Dropping the factor $N$ for a multi-turn coil, giving an answer $N$ times too small.
Using $\mu_0 I/2\pi a$ (straight wire) for the centre of a loop, or vice versa — the loop has no $\pi$ in the denominator.
Forgetting to convert centimetres to metres, which throws the magnitude off by powers of ten.
In the solenoid, plugging in total turns $N$ instead of $n=N/L$ turns per unit length.

How the board asks it

Numericalthe loop and solenoid formulas
A circular coil of $200$ turns and radius $5\ \mathrm{cm}$ carries a current of $2.5\ \mathrm{A}$ . Calculate the magnitude of the magnetic field at the centre of the coil. $(\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}})$
Derive / provethe biot–savart law
Using the Biot–Savart law, derive an expression for the magnetic field at the centre of a circular current-carrying coil of radius $r$ having $N$ turns and carrying a current $I$ .
Numericalvector addition of fields
Two long straight parallel wires, $10\ \mathrm{cm}$ apart, carry currents of $5\ \mathrm{A}$ and $7\ \mathrm{A}$ in the same direction. Find the magnitude of the resultant magnetic field at a point midway between the two wires. $(\mu_0/4\pi = 10^{-7}\ \mathrm{T\,m\,A^{-1}})$
Define / statethe biot–savart law statement
State the Biot–Savart law and write the expression for the magnitude of the magnetic field $dB$ due to a small current element $I\,d\vec{l}$ at a point distant $r$ from it.
Give reasonsfield inside vs outside a toroid
Give a reason why the magnetic field at any point outside an ideal toroid is zero, while inside the core it is $B=\mu_0 N I/(2\pi r)$ , where $r$ is the mean radius.

Practise this topic

NextForce on Moving Charges (Cyclotron & Motion)

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.