Calculating the Magnetic Field Due 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 P is proportional to the current and to the magnitude ds is proportional to sin𝜃 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 curved Wire Segment The magnetic field at O due to the current in the 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 point on AC, ds is perpendicular to hence, 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 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 TWO PARALLEL 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.