Calculating the Magnetic FieldDue to a Current

A wire of arbitrary shape carrying a current i. We want to find the magnetic field B at a nearby point p.

The vector dB is perpendicular both to ds and to the unit vector r directed from ds to P. The magnitude of dB is inversely proportional to r2, where r is the distance from ds to Pis proportional to the current and to the magnitude dsis proportional to sin

Biot–Savart law:

The right-hand rule for determining the direction of the magnetic field surrounding a long, straight wire carrying a current. Note that the magnetic field lines form circles around the wire.

Magnetic Field Due to a straight Wire Segment

Magnetic Field Due to a straight

Magnetic Field Due to a curved Wire Segment

The magnetic field at O due to the current in the curved segment AC is into the page. The contribution to the field at O due to the current in the two straight segments is zero.

The magnetic field at O due to the current inthe straight segments AA’ and CC’ is zero because ds is parallel to along these paths;

Each length element ds along path AC is at the same distance R from O, and the current in each contributes a field element dB directed into the page at O. Furthermore, at every pointon AC, ds is perpendicular to hence,

The direction of B is into the page at O because ds×r is into the page for every length element.

Magnetic Field on the axis of a Circular

Current Loop

Magnetic field lines surrounding a current loop.

THE MAGNETIC FORCE BETWEEN TWOPARALLEL CONDUCTORS

Magnetic Field Outside a Long Straight Wire with Current

Magnetic Field Inside a Long Straight Wire with Current

Magnetic Field of a Solenoid

It concerns the magnetic field produced by the current in a long, tightly wound helical coil of wire. Such a coil is called a solenoid (Fig. 29-17). We assume that the length of the solenoid is much greater than the diameter.

Magnetic Field of a Toroid

• Figure a shows a toroid, which we may describe as a (hollow) solenoid that has been curved until its two ends meet, forming a sort of hollow bracelet.