2. 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:

3. 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.

4. Magnetic Field Due to a straight Wire Segment

5. 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.

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

7. Magnetic Field on the axis of a Circular Current Loop

9. Magnetic field lines surrounding a current loop.

10. THE MAGNETIC FORCE BETWEEN TWO PARALLEL CONDUCTORS

17. Magnetic Field Outside a Long Straight Wire with Current

18. Magnetic Field Inside a Long Straight Wire with Current

22. 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 ). We assume that the length of the solenoid is much greater than the diameter.

26. 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.