Magnetic field due to a long current-carrying wire Oersted's experiments showed that there is a magnetic field around a current-carrying wire. We determine the magnitude of magnetic field at some distance from a long stright wire carrying a current I.

Explanation

The direction of the field is given by the right hand rule. The figure shows an element dl of the current-carrying wire. The point P, where the field is to be determined is at a perpendicular distance 'S' from the wire. The position vector of P from dl is r.

The magnitude dB of the magnetic field due to dl' is given by Biot-Savart law to be
`dB=(mu_(0)I|dl' xxr|)/(4pi r^(2))`
`=(mu_(0)l dl' sin alpha)/(4pi r^(2))`
`=(mu_(0) I dl cos theta)/(4pi r^(2))`
now from the figure `S=r cos theta" which gives "l//r^(2)=cos^(2) theta//s^(2)`
and `l=S tan theta" which gives "dl'=S sec^(2) theta d theta=S d theta//cos^(2)theta`
thus `dB=(mu_(0)l cos theta)/(4pi S) d theta`
we integrate this to get B at P. If the wire is very long then the limits for `theta` would be `-pi//2" to "pi//2`
thus `B=(mu_(0)l)/(2pi S)` (emerging from the paper at P)