biot savort

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S.Y.B.Sc. -II Biot-Savart law N.Kapoor

The magnetic vector potentialMagnetic fields generated by steady currents (and unsteady currents, for that matter) satisfy

(1)

This immediately allows us to write

(2)

since the divergence of a curl is automatically zero. In fact, whenever we come across a solenoidal

vector field in physics we can always write it as the curl of some other vector field. This is not anobviously useful thing to do, however, since it only allows us to replace one vector field by another.Nevertheless, Eq. (2) is one of the most useful equations we shall come across in this lecture

course. The quantity is known as the magnetic vector potential.We know from Helmholtz's theorem that a vector field is fully specified by its divergence and its curl.The curl of the vector potential gives us the magnetic field via Eq. (2). However, the divergence of

has no physical significance. In fact, we are completely free to choose

to be whatever welike. Let

Let us take the curl of Eq. (2). We find that

3

where use has been made of the Coulomb gauge condition. We can combine the above relation

with the field equation to give

The solution of equation is

(4)

The Biot-Savart law

Expression for the magnetic field generated by steady currents by taking the curl of Eq. ( 4). This

gives

But

since is derivative w.r.t. x,y & z, but J(r0) is function of x0 ,y0 & z0 &

Therefore

Equation 5 is known as the Biot-Savart lawafter the French physicists Jean Baptiste Biot and FelixSavart: it completely specifies the magnetic field generated by a steady (but otherwise quitegeneral) distributed current.Let us reduce our distributed current to an idealized zero thickness wire. We can do this by writing

6

where is the vector current (i.e., its direction and magnitude specify the direction and magnitude

of the current) and is an element of length along the wire. Equations 5 and 6 can be combined to

give

7
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S.Y.B.Sc. -II Biot-Savart law N.Kapoorwhich is the form in which the Biot-Savart law is most usually written. This law is to magnetostatics(i.e., the study of magnetic fields generated by steady currents) what Coulomb's law is toelectrostatics (i.e., the study of electric fields generated by stationary charges). Furthermore, it canbe experimentally verified given a set of currents, a compass, a test wire, and a great deal of skilland patience.

Magnetic field due to the current in a straight wire ---

Consider an infinite straight wire, directed along the

-axis, and carrying a current . Let us

reconstruct the magnetic field generated by the wire at point using the Biot-Savart law. Suppose

that the perpendicular distance to the wire is . It is easily seen that

(8)

&

(9)

(10)

Thus, according to Eq. (7), we have

(11)

which gives the familiar result

(12)

So, we have come full circle in our investigation of magnetic fields. Note that the simple result (12)can only be obtained from the Biot-Savart law after some non-trivial algebra. Examination of morecomplicated current distributions using this law invariably leads to lengthy, involved, and extremelyunpleasant calculations.

Magnetic field at the center of a current carrying loop --
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Current Loop

S.Y.B.Sc. -II Biot-Savart law N.Kapoorwe have

integral is from 0 to 2 and since the current is going in the opposite direction so the magnetic field

will be out of the paper

taking the integral gives the magnetic field at the center of the loop

The second more challenging example is the magnetic field at a point z above the loop as shown infigure

Current Loop

The not so obvious hint is the direction of . The cross product of with leads to a vector

perpendicular to both of them and as you go around the loop,

will always be off the z axis by an

angle . This makes all the horizontal components of cancel leaving just the vertical so

once again the differential is given as

, so the integral to get the magnetic field is

From the geometry of the problem we see that

this leads to
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S.Y.B.Sc. -II Biot-Savart law N.Kapoor

substituting these relations into the integral

Finally, taking the integral gives us the magnetic field

Magnetic field due to the current in a Solenoid ---

Solenoid is long wire wound in form of helix such that the length of solenoid is large compared to the radius

of the closely spaced turns.
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biot savort

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