Biot Savart Law in 3D Animation. Oersted Experiment. Class 12. NEET, JEE MAIN.

Biot Savart Law in 3D Animation. Oersted Experiment. Class 12. NEET, JEE MAIN.
On April 21, 1820, Danish Physicist, Hans Christian Oersted, noticed that, a steady current carrying wire, can deflect a nearby magnet. If, the direction of the electric current changed, the magnet, appeared to get deflected, towards the other side. After various tests, and experiments, he summarized that, a steady current carrying wire, produces a steady magnetic field, around it. To determine the direction of the magnetic field, think about holding the wire with a right hand. In this case, the thumb should be, towards the current. This produced magnetic field, will be in the direction, of other four fingers. Thus, Oersted was also able, to show us the direction of the magnetic field.
Still, he couldn't give any idea, about the amount of magnetic field, generated in this procedure.

But, later in this year, two French Physicist, Jean Baptiste Biot, and, Felix Savart, gave the complete Mathematical tool, to calculate the magnetic field. Thus, it is known as the Biot Savart law.

In this animation video, we shall first consider, the scalar form of the Biot savart law. And, thereafter, discuss the full vector concept on this.

To build a realistic concept, on Biot savart law, let us first imagine a wire. It carries a current of amount, I. Also, imagine a point, p, near the wire. All portions of the wire, have their contributions, in generating a magnetic field, at this point, p. A, is an arbitrary point, on the wire. A portion of the wire is taken, about the point, a. This portion is named, dl.

Let us concentrate, on this portion of the wire, dl, about the point, A. Distance between point A, and, point P, is r. We shall now, calculate the magnetic field at point, p, due to this small portion of the wire.

So, let start.
If, the produced magnetic field at p, is dB. Then,

Biot savart law says,
dB is proportional to, I, the amount of current flowing through the wire.

To the length, of the portion of the wire, which is, dl.

To, sine theta, where, theta is angle, formed by, dl and, r. R is the distance, A P.

And, dB is inversely proportional to the, Square of, r.

Summing up all these, we get the complete scalar form, of biot savart law,
Which is, dB equals, mew zero by four pi, into, i, dl, sine theta, by, r square. Mew zero is the magnetic permeability of free space

Here, in this case, the direction of the magnetic field, is not clear. To get its direction, we need to explain the vector form of the law.

And, here it is.

The vector form of this law, is written like this. Note that, there are, arrows, over the letters of, db, dl, and, r. They indicate that, there are three vectors in this equation.

You must be aware that, any vector has two parts. Magnitude, and, direction. On this page, this arrow-head, m, is a vector. Its value, or magnitude is, m. And, cap m, is its direction.

For example, let the gravitational force, acting on a one kg object, be, F. This vector, F, has two parts. First one is its, value, or magnitude. Which is, about 9.8 newton. And the second one is, its direction. Which is downwards. Or, towards minus cap z direction.

After noting all these, the Biot savart law can be written like this. Here, dl, and cap dl, are the magnitude, and, direction of the vector, dl, respectively. And, r, and, cap r, are the magnitude, and direction of the vector, r, respectively.

The direction of dB, is the cross product of, cap dl, and, cap r.

If we calculate, the cross product, we get, sine theta, into cap db. Where, cap db, is the direction of the vector, db. And, it is perpendicular to the plane, that contains vectors, dl, and, r.

So, we understand that, cap db is perpendicular to this plane. But, to which direction? Upwards? Or downward?

To realize, direction of a cross product, imagine a right handed screw driver. Almost, all of us, aware of, a right handed screw driver. On a horizontal plane, if we rotate the screw driver, in clockwise direction, the screw should go downwards. That is, clockwise rotation, yields downwards direction.

Again, if you rotate the driver, in anticlockwise direction, the screw moves upward. Thus, anticlockwise rotation gives upward direction.
In this cross product, we need to rotate, from the first vector, dl, to the second one, r. Off course, through the shorter angular path. And thus, we get its direction, to be downward. Actually, two angles are there. Between the vectors, dl, and, r. One of them is smaller. And a larger another one. We are to move, through the smaller angle. Hence, We get a clockwise rotation here. Thus, the direction of this product vector is, downward.

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