Force on Conductors & Between Wires

PHYMagnetic Field & Forces

A current-carrying conductor in a magnetic field experiences a force $\vec{F}=I\vec{L}\times\vec{B}$ , and two parallel currents exert forces on each other through their mutual fields — the basis of the SI definition of the ampere. ISC numericals ask for the force on a straight wire, the force per unit length between parallel wires (with its attractive/repulsive nature), and current-balance problems where the magnetic force supports a wire's weight.

Direction is set by Fleming's left-hand rule.

Force on a straight current-carrying conductor

F = B I L \sin\theta

L

length of conductor in the field,

\theta

angle between

\vec{L}

(current direction) and

\vec{B}

; maximum when perpendicular (

\theta=90^{\circ}

Force per unit length between parallel wires

\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}

I_1, I_2

the two currents,

d

separation; attractive for same-direction currents, repulsive for opposite.

Current-balance (wire floating against gravity)

B I L = m g \;\Rightarrow\; I = \dfrac{m g}{B L}

m

mass of the wire,

g=9.8\ \mathrm{m\,s^{-2}}

; the upward magnetic force balances the weight

mg

The force is maximum when the conductor is perpendicular to $\vec{B}$ ( $\sin90^{\circ}=1$ ) and zero when parallel.
Direction of force on a conductor follows Fleming's left-hand rule (thumb = force, forefinger = field, middle finger = current).
Parallel wires with currents in the same direction attract; opposite directions repel — the classic exam phrase.
The ampere is defined via $F/L = \mu_0 I_1 I_2/2\pi d$ : $1\ \mathrm{A}$ in two wires $1\ \mathrm{m}$ apart gives $2\times10^{-7}\ \mathrm{N\,m^{-1}}$ .
Convert mass to kg and length to m; a $10\ \mathrm{g}$ wire is $m=10\times10^{-3}=0.01\ \mathrm{kg}$ .
In current-balance problems the equilibrium current is independent of how the field is produced — only $B$ , $L$ and $mg$ matter.
$F=BIL\sin\theta$ is the macroscopic form of $F=qvB\sin\theta$ summed over all the moving charges in the wire.

Where the marks go

Forgetting the factor $2\pi$ in the parallel-wire formula, or writing $d^2$ instead of $d$ in the denominator (the force falls as $1/d$ , not $1/d^2$ ).
Stating that same-direction currents repel — they attract; opposite currents repel.
Leaving mass in grams in the current-balance equation, giving a current $1000\times$ too large.
Omitting $\sin\theta$ when the conductor is not perpendicular to the field.

How the board asks it

Numericalforce per unit length between parallel wires
Two long parallel wires $A$ and $B$ are separated by $0.10\ \mathrm{m}$ and carry currents of $5\ \mathrm{A}$ and $8\ \mathrm{A}$ in the same direction. Calculate the force per unit length between them and state whether it is attractive or repulsive. $(\mu_0=4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}})$
Numericalcurrent-balance (wire supported against gravity)
A horizontal wire of mass $10\ \mathrm{g}$ and length $0.50\ \mathrm{m}$ carries a current and is placed in a uniform magnetic field of $0.40\ \mathrm{T}$ acting perpendicular to it. Calculate the current required so that the magnetic force just supports the weight of the wire. $(g=9.8\ \mathrm{m\,s^{-2}})$
Numericalforce on a straight current-carrying conductor at an angle
A straight conductor of length $0.20\ \mathrm{m}$ carrying a current of $6\ \mathrm{A}$ is held at $30^\circ$ to a uniform magnetic field of $0.50\ \mathrm{T}$ . Find the magnitude of the force acting on the conductor.
Derive / proveforce per unit length between parallel wires; SI definition of the ampere
Two long straight parallel conductors carry currents $I_1$ and $I_2$ in the same direction and are separated by a distance $d$ . Derive an expression for the force per unit length between them and hence explain how this leads to the SI definition of the ampere.
Give reasonsattractive/repulsive nature of parallel currents
Two long parallel wires carry currents in opposite directions. Explain, with reference to the direction of each wire's magnetic field at the other and Fleming's left-hand rule, why the wires repel each other.

Practise this topic

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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.